Exploring the Rate of Convergence of Approximations to the Riemann Integral

Transcription

1 Explorig the Rte of Covergece of Approximtios to the Riem Itegrl Lus Owes My 7, 24 Cotets Itroductio 2. Prelimiries Motivtio Commo Itegrl Approximtios d the Order of Covergece 5 2. The Edpoit Rules The Midpoit Rule The Trpezoid Rule Simpso s Rule Extesios 2 3. Ueve tgs Improper Itegrls Higher Order Error Expsio d the Euler-Mcluri Formul Abstrct There re my well ow wys of pproximtig the vlue of the Riem itegrl of rel-vlued fuctio. The edpoit rules, the midpoit rule, the trpezoid rule d Simpso s rule ech produce sequeces tht coverge to the vlue of the itegrl. Some methods re better th others d this pper sees to qutify how quicly the error of ech pproximtio coverges to zero. After studyig these clssicl results, we exted them i vriety of wys. Sums d products of fuctios with differet tgs d improper itegrls re cosidered, s well s higher order expsio of the error term.

2 Itroductio. Prelimiries We will begi by recordig some defiitios d ottio tht will be used frequetly throughout this pper... The Riem Itegrl The Riem itegrl rises s method of determiig the re uder curve d is cetrl tool i the lysis of fuctios of rel vrible. We will use the defiitios i [3] d the reder is referred to tht source for the proofs of importt theorems. Suppose tht we hve fuctio f : [, b] R. We wt to pproximte the re uder the curve. We re iterested i the siged re, tht is, re bove the x-xis d below the curve is positive re d re uder the x-xis d bove the curve will be egtive re. The followig defiitios will me this ide precise. Fix some lrge umber d divide the itervl [, b] ito subitervls [x, x ], [x, x 2 ],..., [x, x ] so tht x = d x = b. We cll such collectio of subitervls P prtitio of [, b]. These subitervls c hve y legth, but the defiitio mes it cler tht the sum of ll the legths must be equl to b. We cll the legth of the lrgest subitervl the orm of the prtitio, writte P = mx{x x : } For every stisfyig, chose umber t [x, x ]. We cll t tg d the collectio t P of ll pirs of subitervls d their tgs tgged prtitio. Now we observe tht, for ech with, rectgle with bse [x, x ] d height f(x ) hs re roughly equl to the re uder the curve y = f(x) from x = x to x = x. The re of this rectgle is give by f(t )(x x ). Whe we sum these res over ll the subitervls i the prtitio t P, we get pproximtio to the re uder the curve y = f(x) from x = to x = b which we will cll the Riem sum of f with respect to t P, give by S(f, t P ) = f(t )(x x ). As the orm of the tgged prtitio gets smller, we expect tht the Riem sum of f ssocited with tht tgged prtitio coverges to the re uder the curve. This motivtes the followig defiitio Defiitio.. A fuctio f : [, b] R is Riem itegrble o [, b] if there exists umber L such tht for every ɛ > there exists δ > such tht S(f, t P ) L < ɛ for every tgged prtitio t P of [, b] with orm less tht δ. If such L exists, we cll it the Riem itegrl of f from to b d we deote it by f(x) dx or simply f. The bove wy of defiig the itegrl is oly oe pproch. A lterte, d eqully useful, wy of defiig the itegrl, clled the Drboux itegrl, is preseted s follows: Suppose f is bouded fuctio o [, b] d let P = {[x, x ]... [x, x ]} be prtitio of tht itervl. For ech subitervl [x, x ] i P, cosider the two rectgles with shred bse [x, x ] d heights M = sup f(t) d m = t [x,x ] if f(t) t [x,x ] The rectgle with height M is the smllest rectgle with bse [x, x ] d cotiig the curve y = f(x) from x = x to x = x. The rectgle with height m is the lrgest rectgle with tht bse d lyig betwee tht curve d the x-xis. The res of those two rectgles re M (x x ) d m (x x ), respectively. 2

3 If we dd the res of these two sets of rectgles, we get two pproximtios for the itegrl of the fuctio. We will cll these the upper d lower sums of f with respect to P, writte U(f, P ) = M (x x ) d L(f, P ) = It is immedite from the defiitio tht L(f, P ) U(f, P ). m (x x ) for y prtitio P. As the prtitio gets fier, we would expect the upper d lower sums to coverge to the sme umber. This motivtes the followig defiitio. Defiitio.2. A bouded fuctio f : [, b] R is Drboux itegrble if sup L(f, P ) = if U(f, P ) where both the ifimum d the supremum re te over ll prtitios P of [, b]. The commo vlue is clled the Drboux itegrl of f from to b. Proof of the fct tht fuctio is Riem itegrble if d oly if it is Drboux itegrble d the two itegrls hve the sme vlue to [3]. I the remider we will drop the modifiers Riem d Drboux d simply refer to the itegrl of fuctio...2 Asymptotic Nottio I this pper, we will be explorig the log term behvior of vrious sequeces. For sequece tht coverges to limit, either fiite or ifiite, we would lie to be ble to qutify how quicly it coverges to tht vlue d compre its covergece to other sequeces. For exmple, cosider the sequeces whose th terms re l, 2, 2 + 2,! All four of these sequece diverge to ifiity s. However, our ituitio tells us tht some go to ifiity fster th the others. The sequece {l } grows very slowly whe compred to the sequece { 2 } d {!} grows much fster. The sequeces { 2 } d { 2 + 2} both grow t bout the sme rte, s the 2 term begis to domite s gets lrge. These ides bout the reltive rtes of growth of these sequeces re vgue d eed to be mde precise. The ottio itroduced i this sectio llows us to do just this. Defiitio.3. If f d g re rel vlued fuctios d c is rel umber, we sy tht f(x) is of order g(x), writte f(x) = O(g(x)) s x c, if there exist costts δ d M so tht f(x) Mg(x) wheever x c δ. We sy f(x) = O(g(x)) s x if there re costts x d M so tht f(x) Mg(x) wheever x > x. Whe the limit is cler from cotext, we will omit syig s x c. The equls sig i f(x) = O(g(x)) is ot true sttemet of equlity but rther coveiet ottio. This ottio llows us compctly me sttemet bout the limitig behvior of fuctio or sequece. For exmple, cosider the third degree Tylor polyomil of the sie fuctio. We ow from Tylor s theorem tht This is much simpler wy of syig si(x) = x x3 3! + O(x5 ) s x. si(x) = x x3 3! + g(x) where g is fuctio stisfyig g(x) Mx5 for vlues of x sufficietly close to. This covetio will be used throughout the pper d its properties will be further explied whe ecessry. The sttemet f(x) = O(g(x)) mes tht the order of f is, i sese, less th or equl to the order of g i the limit. The ext two defiitios clrify the cses whe the order of f is stricly smller th or equl to the order of g i the limit. 3

4 Defiitio.4. Let f d g be fuctios of rel vrible d let c be exteded rel umber. We sy tht f(x) is of smller order th g(x) s x c, writte f(x) = o(g(x)), if lim x c f(x)/g(x) =. Defiitio.5. If f d g re fuctios of rel vrible d c is exteded rel umber, we sy tht f(x) is symptoticlly equivlet to g(x), writte f(x) g(x), s x c if lim x c f(x)/g(x) =. Returig to our erlier exmple, we c rephrse our clims bout the order of our sequeces usig the ew ottio s follows:.2 Motivtio l = o( 2 ), 2 ( 2 + 2), d = o(!) It follows from our defiitio of the itegrl tht if f : [, b] R is Riem itegrble d { t P } is sequece of tgged prtitios with t P s, the the sequece of Riem sums {S(f, t P )} coverges to f. For exmple, cosider the fuctio f(x) = e x o the itervl [, ]. For y positive iteger, let t P be the prtitio whose subitervls re [( )/, /] with tg / for. This gives us the sequece of Riem sums S(f, t P ) = f ( ) = (e / + e 2/ + + e /) This is ow s the th right edpoit pproximtio of the itegrl of f. To discover its limitig behvior, we strt by observig tht the term iside the pretheses is geometric sum with commo rtio e /, which llows us to write S(f, t P ) = e/ e (+)/ ( e / ) e / = (e ) (e / ) = (e ) ( e / ) By usig the Tylor expsio e x = + x + x 2 /2 + O(x 3 ) s x, we hve The geometric series gives us the expsio which simplifies the bove expressio to [ S(f, t e P ) = ( 2 + O ( )) = (e ) O ( ) 2 S(f, t P ) = (e ) x = + x + x2 + O(x 3 ) s x O = (e ) + e 2 + O ( 2 ( ) 2 ) ( ( + O 2 + O ( )) )] 2 2 This exmple shows tht the itegrl of f over [, ] is equl to e, fct tht we could hve deduced from the fudmetl theorem of clculus. Moreover, we ow hve sese of how quicly the Riem sums coverge to tht vlue. The error term f S(f, t P ) = e 2 coverges to symptoticlly with ( e)/2, tht is f S(f, t P ) e 2 ( ) + O 2 This exmple typifies the geerl situtio. For my methods of estimtig itegrl, we re ble to qutify how quicly those methods coverges to the vlue of the itegrl. This is the gol of Sectio 2 of this pper. 4

5 2 Commo Itegrl Approximtios d the Order of Covergece Suppose tht f : [c, d] R is Riem itegrble. There re my wys to estimte the vlue of d c f. Geerlly, we wt to pproximte the fuctio f by simpler fuctio whose itegrl is esily clculted. The edpoit d midpoit rules pproximte f by rectgle; the trpezoid rule uses, usurprisigly, trpezoids; d Simpso s rule fits qudrtic itersectig f t three poits. The smller the legth of the itervl [c, d] is, the smller the error i this estimte. Suppose tht f is ow defied o some itervl [, b]. Fix lrge vlue of. We will divide [, b] ito subitervls of equl legth d pproximte the vlue of the itegrl over ech subitervl usig oe of our estimtes. I the rest of this sectio we will refer to this itervl divisio by [ I = + b ( ), + b ], d we will deote the edpoits of these itervls by x = + b, Suppose tht A is pproximtio of the itegrl of f over the itervl I, usig oe of the methods we metioed bove. The A = A is our pproximtio for f. We re cocered with the qutity = f(x) dx A which represets the error i this pproximtio. Clerly, we should hve s so tht A coverges to the vlue of the itegrl, but we lso wt to cosider how quicly this umber coverges to. We will exmie this questio i the followig sectios. 2. The Edpoit Rules The simplest wy to pproximte the vlue of itegrl is with the right- d left-edpoit rules. These re respectively give by R = b f(x ), L = b f(x ) Ech of these umbers is simply Riem sum of f usig either the right- or left-edpoits s tgs. The we ow lredy tht L, R f s. The followig theorem chrcterizes how quic this pproximtio coverges. Theorem 2.. Suppose tht the fuctio f hs bouded d itegrble derivtive o [, b]. Let L be the left-edpoit pproximtio to f d let = f L. The we hve f(b) f() (b ) 2 Proof. First, observe tht, by splittig up the itegrl, c be writte s = + = x + = = x f(x) dx b (f(x) f(x )) dx f(x ) Recll itegrble fuctios re lwys bouded. This llows us to defie = s = if f (x) d S x I = sup f (x) x I 5

6 for ech itervl I. The me vlue theorem implies tht Applyig this to our bove formul for, we get = + x A simple computtio shows tht which gives us But we ow tht b 2 s (x x ) f(x) f(x ) S (x x ) s (x x ) dx ( b + x ) = (x x ) dx = = s b 2 b lim s b = lim S = = By the squeeze theorem, it follows tht which estblishes the result = + x (b )2 2 2 ( b lim = b (f(b) f()) 2 S (x x ) dx. ) S. = f (x) dx = f(b) f() A similr result holds for the right edpoit pproximtio. Theorem 2.2. Suppose tht f stisfies the sme hypotheses s the previous theorem. Let R be the right edpoit pproximtio of f d let = f R. The we hve Proof. Observe tht R = L + f() f(b) (b ) 2 f(b) f() (b ) The, usig the previous result, we hve ( b ) b = f R = f L f(b) f() (b ) 2 f() f(b) = (b ) 2 f(b) f() (b ) f(b) f() (b ) The requiremet tht f is differetible d tht its derivtive is itegrble is strog coditio to require of our fuctio. Recll tht fuctio f : [, b] R is sid to be of bouded vritio if V = sup f(x ) f(x < where the supremum is te over ll prtitios [x, x ],..., [x, x ] of [, b]. The umber V is clled the totl vritio of f o [, b]. The followig theorem shows tht if f is of bouded vritio, we c still estblish upper boud o the size of the error of the edpoit pproximtios. 6

7 Theorem 2.3. Suppose tht the fuctio f is of bouded vritio o the itervl [, b] d let V be the totl vritio of f o [, b]. Let be the error f R of the right edpoit pproximtio. The we hve V b Proof. If we split up the itegrl s we did before d perform chge of vribles, we obti = The by the trigle iequlity, we hve = = x (f(x) f(x )) dx (f(x + t) f(x )) dt (f(x + t) f(x )) dt f(x V dt = V b + t) f(x ) dt 2.2 The Midpoit Rule With the left- d right-edpoit pproximtios, we chose the edpoits x d x s the tgs for Riem sum. The ext choice of tg to cosider is the midpoit of ech itervl, (x +x )/2. Specificlly, we hve for ech positive iteger, the midpoit pproximtio M = b = b ( ) x + x f 2 ( f + (2 ) b ) 2 As the followig theorem shows, this turs out to be sigifictly better pproximtio of the itegrl, provided tht we me some stroger ssumptios bout the differetibility of the fuctio. Theorem 2.4. Assume tht f is twice differetible d tht f is itegrble o [, b]. The the error of the midpoit rule stisfies = f(x) dx M f (b) f () 24 2 (b ) 2 Proof. First, set m = (x + x )/2 for. The the qutity we re iterested i becomes = f(x) dx b f(m ) = x [ f(x) f(m ) ] dx Suppose x [x, x ). Becuse f is twice-differetible, we expd f(x) roud f(m ) i secod order Tylor polyomil s f(x) = f(m ) + f (m )(x m ) + 2 f (ζ x )(x m ) 2 7

8 where ζ x is some umber betwee x d m tht depeds o x, d. If we let s = if f (x), x I S = sup f (x) x I the we hve This estblishes x Sice x f (m )(x m ) + 2 s (x m ) 2 f(x) f(m ) f (m )(x m ) + 2 S (x m ) 2 (f (m )(x m ) + 2 s (x m ) ) 2 dx (x m ) dx = d x the bove iequlities reduce to But sice b lim j= by the squeeze theorem, we hve which estblishes the desired result. 2.3 The Trpezoid Rule (x m ) 2 dx = 3 (b ) s j j= b s = lim x [ (x m ) 3] x S = j= x (b ) (f (m )(x m ) + 2 S (x m ) ) 2 dx = S j. j= lim 2 = 24 (b )2 (f (b) f ()) (b )3 2 3 f (x) dx = f (b) f () I the previous two sectios, we estimted the re uder curve by rectgle with specified height. Altertively, we c pproximte the re by trpezoid whose vertices re The re of such trpezoid is give by (x, ), (x, ), (x, f(x )), (x, f(x )). b f(x ) + f(x ) 2 From this, we hve the th trpezoid rule pproximtio of the itegrl of fuctio f give by T = b f(x ) + f(x ) 2 Observe tht the trpezoid rule is the verge of the left- d right-edpoit rules sice T = (R + L )/2. It is ot surprisig, the, tht the trpezoid rule is closer pproximtio to the vlue of the itegrl th either of those two rules o their ow. The followig theorem mes this clim precise. 8

9 Theorem 2.5. If f : [, b] R is twice differetible d T is the th trpezoid pproximtio to the itegrl of f o [, b], the the error term = f T stisfies (b )2 2 2 (f () f (b)) Proof. To reduce clutterig, we will use the ottio tht x = +(b )/ for d h = (b )/ is the orm of the prtitio. Though h d x both deped o, the vlue beig referred to will be idetified by cotext. We begi by fixig vlue of with d estimtig the vlue of the itegrl + x f. We wish to express this itegrl i terms of its vlues t its two edpoits. First we estimte its vlue i terms of its behvior t the left edpoit. For y x, we hve f(x) = f(x ) + (x x )f (x ) + 2 (x x ) 2 f (x ) + O ( (x x ) 3) s x x Whe we me the chge of vribles x x = hs, we get d dx = h ds. This gives us + x f = f(x) + f(x ) + hsf (x ) + 2 h2 s 2 f (x ) + O ( h 3 s 3) ( f(x ) + hsf (x ) + 2 h2 s 2 f (x ) + O ( h 3 s 3)) h ds = hf(x ) + h2 2 f (x ) + h3 6 f (x ) + O ( h 4) s h. Now we use other Tylor expsio, this time of the vlue of f(x + ), tht is, s h. By solvig this equtio for f (x ), we get f(x + ) = f(x ) + hf (x ) + h2 2 f (x ) + O ( h 3) f (x ) = f(x + f(x ) h h 2 f (x ) + O(h 2 ) Substitutig this ito the bove expressio for the itegrl, we get + ( f = hf(x ) + h2 f(x+ ) f(x ) h ) x 2 h 2 f (x ) + O(h 2 ) + h3 6 f (x ) + O ( h 4) = h ( ) f(x ) + f(x + ) h3 2 2 f (x ) + O ( h 4) Whe we sum both sides of this equtio from = to, we get f = T (b )2 2 ( b ) = ( ) f (x ) + O 3 s. The term cotiig the sum is simply the th left-edpoit pproximtio to the itegrl of f, so ( ) b (b lim 2 )2 f T = (f () f (b)) 2 which is equivlet to wht we wted to show. 9

10 2.4 Simpso s Rule So fr, we hve pproximted f with costt d lier fuctios, tht is, degree d degree polyomils. We ow will pproximte f usig qudrtic, tht is, degree two polyomil. A geeric qudrtic polyomil Q(x) hs three degrees of freedom, so we my choose Q so tht it itersect the curve y = f(x) t three poits. We will choose these three poits to be the left- d right-edpoits d the midpoit. If Q(x) is qudrtic fuctio, the we hve d Q(x) dx = c d [ ( ) ] c + d Q(c) + 4Q + Q(d) 3 2 c for y itervl [c, d]. This llows us to defie the th Simpso s rule estimte of the itegrl of f. Defiitio 2.. Let f be Riem itegrble o [, b] d let be eve iteger. For ech i with i, let x i = + i(b )/. The the th Simpso s rule pproximtio to f is S = b (f(x ) + 4f(x ) + 2f(x 2 ) + 4f(x 3 ) + 2f(x 4 ) + 2f(x 2 ) + 4f(x ) + f(x )) I the sme sese tht the trpezoid rule ws the verge of the left- d right-edpoit rules, Simpsos rule is weighted verge of the left-, right- d midpoit rules. Specificlly, S = L + 4M + R 6 The followig lemms d theorem shows tht Simpso s rule is the fstest covergig of ll of the pproximtios we hve cosidered so fr. Lemm 2.. If g hs fourth derivtive o [ r, r] for some positive umber r, the there exists z i ( r, r) such tht r g r r5 (g(r) + 4g() + g( r)) = 3 9 g (z) Proof. Let be the umber r = [ r r 5 g r ] 3 (g(r) + 4g() + g( r)) We wt to show tht there exists z ( r, r) such tht Defie fuctio G o [, r] by G(x) = x r = g (z) 9 The we my compute the first three derivtives of G to be g x (g(x) + 4g() + g( x)) x5 3 G (x) = g(x) + g( x) 3 (g(x) + 4g() + g( x)) x 3 (g (x) g ( x)) 5x 4 = 2 3 (g(x) + g( x)) 4 3 g() x 3 (g (x) g ( x)) 5x 4 G (x) = 2 3 (g (x) g ( x)) 3 (g (x) g (x)) x 3 (g (x) + g ( x)) 2x 3 = 3 (g (x) g ( x)) x 3 (g (x) + g ( x)) 2x 3 G (x) = 3 (g (x) g ( x)) 3 (g (x) g ( x)) x 3 (g (x) g ( x)) 6x 2 = x 3 (g (x) g ( x) + 8x)

11 It is esy to see tht G() = G () = G () = d lso, by our choice of, G (r) =. Becuse G() = G(r), by Rolle s theorem there exists umber (, r) such tht G () =. But the sice G () is lso equl to, there is umber b (, ) such tht G (b) =, gi by Rolle s theorem. Itertig this oe more time gives us c (, b) such tht G (c) =. Substitutig this ito our formul for G gives us c 3 (g (c) g ( c) + 8c) = By solvig for d pplyig the MVT, we fid tht there exists z ( c, c) ( r, r) with This completes the proof. = g (c) g ( c) 9 = g (c) g ( c) = 9 2c 9 g (z) Lemm 2.2. If f hs fourth derivtive o itervl [c, d], the there exists poit v i the itervl (c, d) such tht d f d c c)5 (f(c) + 4f(m) + f(d)) = (d f (4) (v) c where m = (d + c)/2 is the midpoit of the itervl [c, d]. Proof. Let r = (d c)/2 d let g : [ r, r] R be give by g(x) = f(m+x). The g stisfies the hypotheses of Lemm 2 d therefore there exists umber z ( r, r) such tht r r r r5 (g( r) + 4g() + g(r)) = 3 9 g(4) (z) Observe tht which gives us where v = m + z. r r g = d c d c f, g( r) = f(c), g(r) = f(d), r 5 = (d c)5 32 f d c c)5 (f(c) + 4f(m) + f(d)) = (d f (4) (v) Theorem 2.6. If f hs fourth derivtive o [, b] d is eve positive iteger the there exists poit v (, b) such tht (b )5 f S = 8 4 f (4) (v) Proof. Let x i = + i(b )/ for i. Observe tht we my write S = b 3 /2 (f(x 2i 2 ) + 4f(x 2i ) + f(x 2i )) i= The we hve ( / 2i f S = f x 2i 2 i= ) 2(b )/ (f(x 2i 2 ) + 4f(x 2i ) + f(x 2i )) 6 Applyig Lemm 3 for ech iteger i with i /2, we get umber v i (x 2i 2, x 2i ) such tht 2i x 2i 2 f 2(b )/ (d c)5 (f(x 2i 2 ) + 4f(x 2i ) + f(x 2i )) = f (4) (v i )

12 The /2 /2 (d c)5 f S = 9 5 f (4) (d c)5 2 (v i ) = 8 4 f (4) (v i ) Recll tht derivtives stisfy the itermedite vlue property. If we iterpret i= f (4) (v ) + f (4) (v 2 ) + + f (4) (v /2 ) /2 s verge, the it must lie betwee mi{f (4) (v i ) : i /2} d mx{f (4) (v i ) : i /2}. Sice f (4) hs the IVP o [v, v /2 ], there is v (v, v / ) such tht i= which is wht we wted. f S = (b )5 8 4 f (4) (v) 3 Extesios 3. Ueve tgs We will begi by extedig the right- d left-edpoit rules. These rules choose fixed tgs i uiform prtitio of the itervl [, b]. The right edpoit rule chooses the poit x i the itervl [x, x ] s the tg, while the left-edpoit rule chooses the poit x. Wht if isted we choose poit fixed distce from oe of the edpoits i ech subitervl? The ext theorem swers this questio. For the remider of this sectio, we will cosider oly fuctios o the itervl [, ]. Results for geeric itervls [, b] exist d re esy to deduce from the followig theorems by cosiderig the trsformtio x (x )/(b ) from [, b] to [, ]. Theorem 3.. Suppose the fuctio f : [, ] R is differetible d its derivtive is itegrble. For y α [, ], form the Riem sum U = ( ) α f The the error term = f U stisfies f() f() (α /2) Proof. The result follows by observig tht = f ( ) α f ( = f ( ) ) f + ( f ( ) f ( )) α By the me vlue theorem, for ech with, there exists vlue c [( )/, /] such tht ( ) ( ) α f f = α f (c ). The we hve ( = f f() f() 2 ( ) ) f + α + α f, f (c ) 2

13 here we hve used Theorem 2.2 d the fct tht f (c ) is Riem sum of f over [, ]. Applyig the fudmetl theorem of clculus gives us 2 (f() f()) + α (α /2)(f() f()) (f() f()) = We c observe tht whe we choose α = d α =, we get the right- d left-edpoit pproximtios to the itegrl d the theorem gives us the expected symptotic behvior from Thereoms 2. d 2.2. Oe might thi tht we could hve just prove this theorem isted of goig to the trouble of provig Theorems 2. d 2.2, but remember tht we use Theorem 2.2 i the proof of this theorem. It is lso worth observig tht if we left α = /2, we get the midpoit pproximtio of the itegrl of f. Theorem 3. tells us, correctly, tht f M = o(/), tht is, the order of covergece of the error term is strictly smller th /. The midpoit rule (Theorem 2.4) clrifies tht it is i fct symptotic with C/ 2, for costt C. We ow further geerlize by sig the questio: Wht hppes if we form Riem sum pproximtig the sum or product of two fuctios, sy f + g or fg, but we evlute f d g t differet tgs? More precisely, suppose tht for every iteger with, we pic tgs x, y [( )/, /] d form the sums S = (f(x ) + g(y)) d P = f(x )g(y) It is cler by lierity tht S f s. The followig theorem shows tht P lso coverges to wht we would expect. This result is commoly ow s Bliss theorem. Theorem 3.2. Let f, g : [, b] R be Riem itegrble o [, b] d suppose tht x, y [( )/, /] for. Let P = f(x )g(y). The we hve lim S = f. Proof. Fix y ɛ >. Sice f d g re itegrble, their product fg is lso itegrble. The, there exists iteger N so tht f(y )g(y) f(x)g(x) dx < ɛ wheever > N. Further, by the Cuchy criterio, we my choose possibly lrger N so tht ( g(x ) g(y) ) < ɛ s well wheever > N. Sice f is Riem itegrble, it is bouded. Let M = sup x [,b] f. The 3

14 wheever > N, we hve P f(x) dx = f(x )g(y) f(x) dx f(x )g(y) f(x )g(x ) + f(x )g(x ) f(x) dx [ f(x ) g(x ) g(y)] + ɛ M g(x ) g(y) + ɛ < ( + M)ɛ This shows tht P fg s, which is equivlet to wht we wted to show. Wht c we sy bout the symptotic behvior of the error of ech of these pproximtios? If the distce from ech edpoit is fixed, s it ws i Theorem 3., we re ble to qutify this behvior. For the sum, the swer is provided to us esily by Theorem 3.. Suppose tht α d β re fixed umbers i the itervl [, ]. The (f + g) ( ( ) ( )) ( α β f + g = f ( ) ) ( α f + g ( ) ) β g ((α /2) (f() f()) + (β /2)(g() g())) The cse for products is slightly more complicted d ws cosidered i the pper []. We reproduce their result for this situtio, with slightly modified proof. Theorem 3.3. Let f, g : [, ] R be two differetible fuctios whose derivtives re Riem itegrble. Let α, β [, ] d let = for every positive iteger. The we hve Proof. Let lim f()g() f()g() = + α 2 D = f(x)g(x) dx ( ) ( ) α β f g f(x)g(x) dx Sice f g is cotiuous d therefore itegrble, we hve f (x)g(x) dx + β f ( ) g lim D f()g() f()g() = 2 ( ). f(x)g (x) dx by Theorem 2.. Becuse f is differetible, the me vlue theorem implies tht for ll itegers d with, there re umbers c, d with α < c <, β 4 < d <

15 d The we hve f ( ) f which by Theorem 3.2 coverges to ( ) α = f (c ) α (, g ) g ( ) β = g (d ) β ( ) ( ) α β = f(x)g(x) dx f g [ ( ) ( ) ( ) ( )] α β = D + f g f g [ ( ) ( )] ( ) α = D + f f g ( ) [ ( ) ( )] α β + f g g = D + α ( ) ( ) α f (c ) g + f g (d ) f()g() f()g() 2 + α f (x)g(x) dx + β f(x)g (x) dx 3.2 Improper Itegrls We ow tur our ttetio to improper itegrls. Recll tht fuctio f : (, b] R hs improper itegrl if c f exists for ll c (, b) d lim c + f exists d is fiite. A similr defiitio exists c for itervls of the form [, b) d (, b). For exmple, if < r <, the fuctio f(x) = /x r o (, ] is improperly itegrble s we c see by lim δ + δ f = lim δ + r ( δ r ) = r We ow s, our the pproximtios tht we hve bee usig for the itegrl of fuctio still vlid for fuctios tht re improperly itegrble, but ot Riem itegrble. For exmple, if f : (, ] R is improperly itegrble, we c form the sum R = ( ) f We plce the tilde over the R to remid us tht this is differet type of qutity th the right-edpoit pproximtio from Sectio., sice we do ot ow tht f is Riem itegrble. The ext theorem shows tht, uder some coditios, this fuctio does, i fct, coverge to the vlue of the itegrl. Theorem 3.4. Let f : (, ] R be o-icresig d improperly itegrble d let R be s bove. The R f s. Proof. Fix y ɛ >. Sice c f f s c +, we c fid umber > such tht f = f f < ɛ. For every positive iteger, let j be the uique iteger such tht j / < (j + )/. The R is right-edpoit pproximtio of j/ f, so we my fid vlue N such tht j/ f R < ɛ wheever > N. Observe tht sice f is decresig, we hve j f(/) < 5 j/ f.

16 The wheever > N, f R f + f R j ɛ + f(/) + f f(/) j / =j + 4ɛ. The it follows tht R f s. j f The ides used i the bove proof c esily be exteded to more geerl situtios, tht is for fuctios o geerl itervls (, b) d for the left edpoit method s well. Wht c be sid bout the order of covergece of this pproximtio? The uthor is ot wre of geerl swer to this questio d it ppers to be more complicted tht the cse of Riem itegrble fuctios. We provide exmple to illustrte the situtio Theorem 3.5. Suppose tht < r < d let g : (, ] R be give by g(x) = /x r. The there exists o-egtive costt C such tht g R C r where R is the sme s i the previous theorem. To prove this theorem, we eed the followig lemm Lemm 3.. Suppose tht f : [, ) R is o-egtive differetible fuctio whose derivtive is egtive d icresig o (, ). For ech positive iteger, let A be the bsolute vlue of the error of sigle subdivisio trpezoid pproximtio of the itegrl of f o [, + ]. The the series A coverges. Proof. First, for ech positive iteger, defie the fuctios L (x) = (f() f( ))(x ) + f(), U (x) = (f( + ) f())(x ) + f() Observe tht f is icresig therefore f is covex, which implies tht L (x) f(x) U (x) for ll x [, + ]. Also ote tht, sice U (x) is sect lie coectig (, f()) d +, f( + ), so A = + (U f) If we let D = f( + ) f() for 2, simple computtio shows tht from which we get A = + + (U L ) = 2 (D D )) (U f) + (U L ) = 2 (D D ) Observe tht sice f is egtive d icresig, it coverges to fiite limit s x. Therefore, by the me vlue theorem, D = f( + ) f() coverges to fiite limit s. The telescopig series =2 D coverges d the smller series A coverges s well. We ow hve the tools to prove Theorem

17 Proof. We wt to show tht the sequece r ( ) dx x r R = r r r coverges. Let f(x) = /x r for x <. The by pplyig Lemm 3.2 to f, we tht get A = dx x r f() + (f() + f()) 2 = r r x r + O() coverges. Therefore the sequece we re iterested i coverges s well d the result is prove. 3.3 Higher Order Error Expsio d the Euler-Mcluri Formul So fr, the results we hve bout the order of covergece of pproximtio to the Riem itegrl hve looed lie C p for some costt C d d some iteger p. Aother wy of writig this would be = C ( ) p + o p. We might hope to dd some more terms to this symptotic expsio. A geerl symptotic expsio of the error term would loo lie = C + C C ( ) + o for some fixed positive iteger. To ccomplish this gol, we must itroduce power tool tht is used to coect sums d itegrls clled the Euler-Mcluri summtio formul. We will drw this dicussio from [4] d dopt tht pper s ottio. We begi by defiig the Beroulli umbers d polyomils, which hve my useful properties. We defie the th Beroulli polyomil recursively, for ech iteger by B (x) =, B (x) = B (x), d B (t) dt = for ll vlues of x. Note tht the polyomils re well-defied, sice we hve B (x) = B (t) dt + B for some choice of B. There is uique choice of B tht mes the bove equtio hold. We cll B ( ) the th Beroulli umber. Obeserve tht B = B () for. For lrger th 2, we hve dditiolly B () = B () + (B () B ()) = B () + The first severl Beroulli polyomils d umbers re B = B () = B. 7

18 B (x) = B = B (x) = x 2 B = 2 B 2 (x) = x 2 x + 6 B 2 = 6 B 3 (x) = x x2 + 2 x B 3 = B 4 (x) = x 4 2x 3 + x 2 3 B 4 = 3 B 5 (x) = x x x3 6 x B 5 = B 6 (x) = x 6 3x x4 2 x B 6 = 42 For every positive iteger, we my defie the th periodic Beroulli polyomil by P (x) = B (x x ) for ll rel umbers R. Sice B () = B () for 2, P is cotiuous for 2. Usig these ew tools, we ow stte d prove the Euler-Mcluri summtio formul. Theorem 3.6. Suppose f : [, ] R is m + times cotiuously differetible. The we hve f = f() m 2 (f() + f()) + ( ) j B j j! = + ( ) m+ (m + )! j= ( ) f (j ) () f (j ) () f (m+) (x)p m (x) dx where B is the th Beroulli umber d P is the th periodic Beroulli polyomil. Proof. We will procede by iductio o m. We begi with the bse cse m =. Fix umber with. The recllig tht P (x) = for ll x, we hve f = f(x)p (x) dx. By performig itegrtio by prts with we get u = f(x) dv = P (x) dx du = f (x) dx v = P (x) = x 2 [ ] f = f(x)p (x) f (x)p (x) dx = 2 (f( ) + f()) f (x)p (x) dx We ow perform itegrtio by prts gi o f (x)p (x) dx with to get The we hve u = f (x) du = f (x) dx dv = P (x) dx v = P 2 (x)/2 f (x)p (x) dx = B 2 2 (f () f ( )) f (x)p 2 (x) dx. 2 f = 2 (f( ) + f()) B 2 2 (f () f ( )) + f (x)p 2 (x) dx 2 which we my sum from = to to get the m = 2 bse cse. Now fix some positive iteger m d suppose tht the mth cse of our propositio holds. Cosider the remider term m! f (m) (x)p m (x) dx We gi cosider oly the prt of the itegrl o [, ] d perform itegrtio by prts with 8

19 to get m! u = f (m) (x) dv = P m (x) dx du = f (m+) (x) dx v = P m+ (x)/(m + ) f (m) (x)p m (x) dx = B m+ (m + ) (f (m) () f (m) ( )) (m + )! f (m+) (x)p m+ (x) dx Summig this from = to d substitutig it for the error term i the mth cse yields the (m + )th cse, completig the proof. This formul llows us to exted our symptotic expsio of the error term of our itegrl pproximtio s fr out s we hve derivtives. For exmple, suppose tht f : [, ] R is seve times differetible. The Euler Mcluri formul gives us where f f( ) + f() 2 = 7 ( ) j B j j! (f j () f j ()) j=2 = 2 (f () f ()) 2 (f () f ()) ! (f (5) () f (5) ()) + ρ ρ 7! f (7) (x) P 7 (x) dx 7! f (7) (x) dx where we hve used the fct tht P 7 (x) for ll x. Suppose ow tht g : [, ] R is seve times differetible. We wt to expd the error term g T usig our ew formul. By lettig f(x) = g(x) i the bove formul, we hve f T = 2 2 (f () f ()) 2 4 (f () f ()) ! (f (5) () f (5) ()) + O(/ 7 ) which is much more precise expsio th our origil Acowledgmets f T 2 2 (f () f ()) I would lie to th Russ Gordo for his help d ptiece with the writig of this pper. I would lso lie to th Albert Schueller for orgizig this project d Jmie Ediso d Ev Kleier for providig edits d feedbc throughout the semester Refereces [] Meriescu, D. & Moe, M. (2) A extesio of result bout the order of covergece Bulleti of Mthemticl Alysis d Applictios Vol. 3, Iss. 3, [2] Poly, G. & Szego, G. (972) Problems d theorems i lysis. Berli: Spriger. [3] Gordo, R. (22) Rel lysis: first course. Addiso-Wesley Higher Mthemtics [4] Apostol, T. (999) A Elemetry View of Euler s Summtio Formul Americ Mthemticl Mothly Vol. 6, No. 5 [5] Rudi, W. (986) Rel d Complex Alysis. 3rd Ed. McGrw-Hill Sciece/Egieerig/Mth - 9