Archivum Mathematicum

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1 Archivum Mthemticum Erich Brvínek On some convergence tests for series nd integrls Archivum Mthemticum, Vol. 3 (1967), No. 2, Persistent URL: Terms of use: Msryk University, 1967 Institute of Mthemtics of the Acdemy of Sciences of the Czech Repulic provides ccess to digitized documents strictly for personl use. Ech copy of ny prt of this document must contin these Terms of use. This pper hs een digitized, optimized for electronic delivery nd stmped with digitl signture within the project DML-CZ: The Czech Digitl Mthemtics Lirry

2 55 ON SOME CONVERGENCE TESTS FOR SERIES AND INTEGRALS E. BARVINEK (BRNO) Received Novemer 31, For determining convergence of series ]T n 6 n the four following w=-0 theorems usully re pplied, see [1]. [AB-J Ael's test. The series S n n converges if S n converges nd if ( n ) is monotonic convergent sequence. [DR X ] Dirichlet's test. Let S n e series whose prtil sums form ounded sequence. Let ( n ) e monotonic sequence which converges to 0. Then S n n converges. [DBRJ Du Bois-Reymond's test. If oth series S n nd S n 6 n+1 converge then S n n conveges too. [DDJ Dedekind's test. Let S n e series whose prtil sums form ounded sequence. Let S n n+l \ converge nd (6 n ) converges to O. Then S n 6 n lso converges. Let f(x) nd g(x) e functions defined in n intervl [, 6[ on the ' set of rel numers. Let the proper Riemnn integrls f(x) dx nd f g(x) dx exist for ny ' e [, 6[. n For determining convergence of the improper integrl f(x) g(x) dx the two folowing theorems usully re pplied. [AB 2 ] Ael's test. If the integrl f(x) dx converges nd g(x) is monotonic nd ounded on [, 6[, then f(x) g(x) dx converges. ' [DR 2 ] Dirichlet's test. If there is f(x) dx \ <J M for some con stnt M nd ny 6' e [, 6[, g(x) is monotonic on [, [ nd lim g(x) 0, x-^ then f f(x) g(x) dx converges. We wish to get two theorems on improper integrls which would correspond to the theorems [DBRJ nd [DD X ] out series.

3 56 2. The proofs of the theorems [AB 2 ] nd [DR 2 ] re sed on the Second Men Vlue theorem for Riemnn integrls. For our purpose we would need ny men vlue theorem in which no monotonic ehviour is supposed. Let us mention oth clssicl theorems. [L x ] First Men Vlue theorem. Let proper Riemnn integrls ' f f(x) dx nd f g(x) dx exist. Let g(x) = 0 nd m _ f(x) g M on [, '], where m, M re rel numers. Then there exists rel numer V ' s e [m, M] such tht f f(x) g(x) dx = s f g(x) dx. [L 2 ] Second Men Vlue theorem. Let proper Riemnn integrl ' f f(x) dx exist. Let g(x) e monotonic on [, ']. Then there exists h' s s e [, '] such tht f f(x) g(x) dx = g() f f(x) dx + g () f f(x) dx. ft The two theorems we re giving elow lso function s men vlue theorems. ' ' [L 3 ] Let f f(x) dx nd g(x) dx exist nd g(x) ^ 0 on [, ']. Then ' there exists s e [, '] such tht f f(x) g(x) dx = sup f(x) f g(x) dx + xe[,'] ' + inf f(x) f g(x) dx. xe[,'] s Proof. Let us put M = sup f(x), m = inf f(x). For the continuous x xe[,'] xe[,'] function F(x) = f (M m) g(t) dt f f(t) m) g(t) dt there ' s holds F() g 0, F(') ^ 0. Hence there is numer s e [, '] such tht F(s) = 0. [LJ Let f(x) nd g'(x) = I - J e Riemnn integrle on ' ' s ' [, ']. Then (1) f f(x) g(x) dx = (g'(s) ff(x) dx) ds + g(') f f(x) dx. Proof. Let us compute the derivtion t n ritrrily fixed xe], '[ X 8 x of the function F(x) = f (g'(s) f f(t) dt + f(s) g(s)) ds g(x) f f(s) ds ccording to the definition of derivtion. We hve F(x + h) F(x) = x+h H x+h -= (9'(s) f f(t) dt + f(s) g(s)) ds - (g(x + h) g(x)) f f(s) ds x

4 f X+Һ x+h x+h g(x)f (*) d s = f (9'W If(t)dt+f(s)9( s ))^~f 9'{s) dsff(s)ds x x x x+h x+h * x y l g{x) f f(s) d = [?'(«) f f(t) dt +(«) (t7(«),?(.-))] ds = j f(t)dt+f(s)fg'(t)dt)ds. x+h * *-f h 57 (gr'(«) We cn suppose tht («) g M nd g (s) g M on [, 6']. Becuse s lies etween x nd + h we otin F(x + h) F(x) \^\h\(m\h\ M + M \h\m) = 2Jf 2 i 2. Hence F'(x) = 0. Thus F(z) is constnt on K VI From the continuity of F(x) on [, 6'] there follows tht for x e [, '] there is F(x) = F() = 0. Putting x = ft' we otin the formul of the theorem. Theorem [L 4 ] presents integrtion y prts, see [3], nd implies the formul of [L 2 ] provided moreover g(x) is monotonic, see [2]. E.g; if g(x) is nondecresing, it is g'(x) ^ 0 nd [L-J cn e pplied to the ' * *o ' integrl g'(s) f f(x) dxds. We get f(x) dx f g'(s) ds = (g( f ) *o 9( )) f f( x ) dx where s 0 is convenient numer in [, ']. Hence the formul of [L 2 ] follows. We do not give here ny ppliction of [L 3 ], wheres [L 4 ] will e pplied to otin the following theorems [DBR 2 ] nd [DD 2 ]. 3. Let f(x) nd g(x) e functions defined in n intervl [, [ on the ' set of rel numers. Let the proper Riemnn integrls f(x) dx nd ' f g'(x) dx exist for ny ' e [, [, where g'(x) dg(x) dx [DBR 2 ] If the integrls f(x) dx nd g'(x) \ dx converge then ь the integrl f(x) g(x) dx converges. Proof, For ritrry ' e [, [ there holds formul (1). For ' -> ~ v 8 the integrl (g'(s) f f(x) dx) ds converges solutely, ecuse the s function f f(x) dx of the vrile s is ounded nd the integrl g'(s) ds converges solutely. As to the term #') (#) dx, there exists lim V->~ ' '

5 58 g(') = g() + f g'(s) ds nd hence finite limit of g(') f f(x) dx exists. [DD 2 ] ' Let f f(x) dx ^ M for every ' e [, [. Let the integrl f g'(x) \ dx converges nd lim g(x) = 0. Then the integrl f f(x) g(x) dx x-+~ converges. Proof. For ll 'e[, [ there holds formul (1). For ' -> ~ the ' s integrl J (g'(s) f f(x) dx) ds converges for the sme resons s in ' [DBR 2 ] nd g(') f f(x) dx converges to For i = 1 or i = 2 there re certin reltions mong the theorems [AB ( ], [DB { ], (DBB t ], [DD { ], see [4]. [DG] For i = 1,2 there holds the impliction-digrm [DD,] => [DBR { ] V V [DP,] => [AB { ] Proof, ) [DP X ] => [AB X ]. Under the suppositions of [AB X ] there exists finite limit lim n =. Let us write n n = n + n ( n ). n >oo N The series 2 n ( n ) converges ccording to [DR X ], ecuse n ^ n=0 _ M nd ( n ) converges to 0. ) [DP 2 ] => [AB 2 ]. Under the ssumptions of [AB 2 ] there exists finite limit lim g(x) = s. Let us write(.r) g(x) = sf(x) -f f(x) (g(x) s). x^ The integrl f f(x) (g(x) s) dx converges ccording to [DP 2 ]> ecuse ' f f(x)dx?g M nd g(x) s converges monotoniclly to 0 if x -> \ c) [DBR X ] => [AB X ]. Under the ssumptions of [AB X ] the series S(6 M & n +i) converges solutely. d) [DBR 2 ] => [AB 2 ], Under the ssumtions of [AB 2 ] finite limit lim g(x) = e exists, Let u$ put ^ = 1 or 2 = 1 ccording to g(x) '

6 r eing nondecresing or nonincresing. Then J g'( x ) \dx = z f g' (x) dx = «= z(g(') g()). Hence the integrl g'(x) d; converges. e) [DD X ] => [DR X ]. Under the ssumptions of [DR X ] the convergence of the series 2 n 6 n+1 follows. f) [DD 2 ] => [DR 2 ]. Under the ssumptions of [DR 2 ] the integrl J I 9( x ) I d# converges. g) [DDJ => [DBR X ]. Under the ssumptions of [DBR X ] from the convergence of the series 2(6 n 6 n+1 ) the existence of finite limit lim n = n»oo = 6 follows. Let us put n n = n + n ( n 6). The series 2 n ( n 6) n converges ccording to [DDJ ecuse v \ ^ M, nd 2 ( n 6) (6 M+1 6) = 2 6 n 6 W+1 converges. The sequence (6 n 6) converges to 0. h) [DD 2 ] => [DBR 2 ]. Under the ssumptions of [DBR 2 ] the existence of finite limit lim g(') = s follows from the convergence of the integrl '-+~ f g'(x)dx. Let us put f(x) g(x) = sf(x) + f(x) (g(x) s). The integrl ff( x ) (9( x ) S) d.r converges ccording to [DD 2 ] ecuse f(x) dx \ g M, the integrl (g(x) 5)' dx = g'(x) dx converges nd lim (g(x) s) = 0 x->~ 5. For the uniform convergence of the series 2 n (y) n (y), y e Y, where Y is n ritrry set of rel numers, the tests nlogous to [AB X ], [DR X ], [DBR X ], [DD X ] re used. For uniform convergence of improper Riemnn integrls f(x, y) g(x, y) dx, y e Y Ael's nd Dirich let's tests [AB±] nd [DR 4 ] re used. Suppose tht the functions f(x, y) nd g(x, y) re defined for x e [, 6[ ' ' exist for ny ' e [, [ nd y e Y. ' ' 59 C do(x 11} f(x, y) dx nd l -r-- d#

7 60 [AB 4 ] Ael's test. If the integrl f f(x, y) dx converges uniformly for y e Y, \ g(x, y) ^ M for x e [, [ nd y e Y, nd for ny y e Y the function g(x, y) of the vrile x is monotonic in [, [, then f f( x * y) 9( x > y) dx converges uniformly for y e Y. ' [DRJ Dirichlet's test. If f f(x, y) dx < M for ny ' e [, [ nd y e Y, lim g(x, y) -= 0 uniformly on Y nd g(x, y) s function of x z-+~ is monotonic in [, [ for ny y e Y, then f f(x, y) g(x, y) dx converges uniformly on Y. Below we give two tests for uniform convergence of improper Riemnn integrls nlogous to [DBR 2 ] nd [DD 2 ]. [L 5 ] Let f(x) nd g'(x) e integrle in [, '] for ny ' e [, [. Let the integrl f f(x) dx converge. Then for every ', " e [, [ there holds the formul " " (2) ff(x)g(x)dx=f(g'(s)ff(t)dt)ds-g(")ff(t)dt + g(')ff(t)dt. ' ' s " ' Proof. Applying formul (1) for the intervls [, '] nd [, "] " nd sutrcting we get formul contining f f(t) dt, f f(t) dt nd V s' " ' f f(t) dt. For s' == s, ", ' let us put f f(t) dt = f f(t) dt f f(t) dt. s' Then we otin (2). [L 6 ] Assume tht the integrl f f(x, y) dx converges for ny y e Y, " tht for ny " > \ f ->~ the integrl f ( 9^ y- ff(t,y)dt\ds converges for y e Y uniformly to 0, nd for ' -> ~ the function g(\ y) ff(t>y)dt converges for y e Y uniformly to 0. Then the integrl ' f f( x > y) 9( x > y) dx converges uniformly on Y. Proof. We re to show tht for " > V, ' -> 6~ the integrl ' s

8 61 ' " J i( x > y) 9( x > y) dx converges uniformly to 0 on Y. For ny y e Y ccorv " " ding to (2) we hve f f(x, y) g(x, y) dx = f (^j^ [f(t, y) dt\ ds ' ' s ~~~ <](\ y) f f(t, y) dt + g(', y) f f(t, y) dt. All terms on the right " ' converge uniformly to 0 on Y whenever 6" > ', ' -> 6~. C C 3o(x v\ [DHR 4 ]. Let the integrls f(x, y) dx nd ^~~ dx converge J J I dx I uniformly on Y. If g(x, y) \ <Z M for x e [, [ nd y e Y, then J f( x > y) g( x >y) dx converges uniformly on Y. " Proof. The integrl J ~^~ f(t, y) dt\ ds converges uniformly ' to 0 on Y whenever " > ', ' -> ~, ecuse for s -> 6~ the integrl ь f f(t> y) dt converges uniformly to 0 nd thus there exists constnt K,s such tht f f(t, y)dt\ <* K for x e [, [ nd y e Y. Hence 8 \f(^f JM^iK f]^ ' s ' nd the right-hnd side converges uniformly to 0. The ffirmtion follows from [E 6 ]. ' [DD 4 ] Let ff(x, y)dx\ <i M for ' e [, 6[, y e Y. Let the integrl i ' I Sg(x, y) ôx dx converge uniformly on Y nd let e lim g(x, y) == 0 x-+~ uniformly on Y. Then integrl f(x, y) g(x, y) dx converges uniformly on Y, Proof. For ny y e Y nd ny ' e [, [ we cn pply [D 4 ] nd write ccording to (1)

9 62 v Jf(x,y,)g{x,y)âx = - J - 9 -^- Jf(t,y)dř)d* + g(', y) Jf(x,y)d.s. The first term on the right converges uniformly ecuse f f(t, y) dt ^ < M nd - ' - I d. converges uniformly. The second term s converges uniformly to 0. As to the impliction-digrm for the uniform convergence of integrls we cn prove only the following propositions. Anlogous results my e otined for series. [Pi] I f I M y)^x I.S K for y e Y, then [DDJ => [DBRJ. Proof. Under the suppositions of [DBRJ there exists uniform limit lim g(x, y) = s(y) nd there is s(y) \ g M. Let us wtite f f(x, y) x-+~ g(x, y) dx = s(y) f f(x, y) dx + f f(x, y) (g(x, y) s(y)) dx. The first term on the right converges uniformly ecuse s(y) is ounded. The second term converges uniformly ccording to [DDJ ecuse ' I f f(xj y)dx\ ^ K' for y e Y nd every ' ner, the integrl! A I (g(x, y) s(y)) I dx converges uniformly nd there is uniformly I ox ' I on Y lim (g(x, y) s(y)) = 0. x->~ [PJ It holds [DDJ ^ [DRJ. Proof. Under the ssumptions of [DRJ let us put z(y) = 1 or z(y) = = 1 for ny y e Y ccording to g(x, y) eing nondecresing or ' ' OQ(X 1\ i -^- - I d:! 3Q(X u) C BQ[X H\ ~- dx = z(y) I ^--^dx = = z (y) l9(', y) 9(,y)] which converges uniformly to the limit z (y) g( > y). Hence the ssumptions of [DDJ re fulfilled.

10 [P 3 ] Let Y e ounded nd closed set of rel numers. Let g(x, y) e continuous on Y for ny x e [, [ nd s(y) == lim g(x, y) e contix-+~ nuous on Y. Then [DBR 4 ] => [ABJ. Proof. Under the ssumptions of [AB 4 ] there exists finite limit lim g(x, y) = s(y) for ny y e Y ecuse g(x, y) is monotonic in x for x~>ny y e Y nd g(x, y) \ l M. Let z(y) e defined like in [P 2 ]. The ' c)o(x?) ^- dx = z(y) (g( f, y) g(, y)] converges uniformly if nd only if g(x, y) -> s(y) does so. This follows from DINI'S theorem. [P 4 ] Let \ff(x,.y) dx\ g K for y e Y where Y is ounded nd closed set of rel numers. Let g(x, y) e continuous on Y for ny x e [, [ nd s(y) = lim g(#, y) e lso continuous on Y. Then [DR 4 ] =>.r->6- => [ABJ. Proof. Under the suppositions of [AB 4 ] there exists finite limit lim g(x, y) = 5(2) for ny y e Y nd s(y) 5j M. Let us write f f(x, y) x->~ g(x, y) dx) = s(y) f f(x, y) dx + f f(x, y) (g(x, y) s(y)) dx. The first term on the right converges uniformly. So does the second term ccording ' to [DRJ ecuse for ' ner nd y e Y there is f(x, y) dx ^ K' nd lim [g(x, y) s(y)] == 0 on Y uniformly y Dini's theorem. x-+~ 63 REFERENCES [1] K. Knopp, Theorie und Anwendung der unendlichen Reihen, B rlin 1931 [2] K. Kurtowski, Wykldy rchunku rozniczkowego i clkowego I Wгszw 1948 [3] V. Jrník, Úvod do počtu integrálního, Prh 1948 [4] G. M. Ficht ngolc, Kur8 differenciálnogo i integrlnogo iәčislenij I, Moskv 1959

Chapter 6. Riemann Integral

Chapter 6. Riemann Integral Introduction to Riemnn integrl Chpter 6. Riemnn Integrl Won-Kwng Prk Deprtment of Mthemtics, The College of Nturl Sciences Kookmin University Second semester, 2015 1 / 41 Introduction to Riemnn integrl