6. I njinite Series j. To be specific we shall indicate a rearrangement of the series 1-!-!-!-! !.

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1 . Functions of a Real Variable I. A divergent series whose general term approaches zero. The harmonic series I)/n. 2. A convergent series La,. and a divergent series Lb n such that a" bn, n = I, 2, '". Let an == 0 and b" == -/n, n =, 2, '". 3. A convergent series Lan and a divergent series Lb n such that I a" I ~ I b" I, n = I, 2, '". Let La" be the conditionally convergent alternating harmonic series L( -)n+l/n, and let Lb.. be the divergent harmonic senes Ll/n. 4. For an arbitrary given positive series, either a dominated divergent series or a dominating convergent series. A nonnegative series La,. is said to dominate a series Lb n iff an ~ let an I b", I for n =, 2,... If the given positive series is Lb", b" for n =, 2, '". Then if Lb n diverges it dominates the divergent series La,., and if Lb" converges it is dominated by the convergent series Lan' The domination inequalities can all be made strict by means of factors! and 2. This simple result can be framed as follows: There exists no positive series that can serve simultaneously as a comparison test series for convergence and as a comparison test series for divergence. (Cf. Example 9, below.) 5. A convergent series with a divergent rearrangement. With any conditionally convergent series Lan, such as the alternating harmonic series L( -l),,+l/n, the terms can be rearranged in such a way that the new series is convergent to any given sum, or is divergent. Divergent rearrangements can be found SO that the sequence (8,.j of partial sums has the limit + 00, the limit - 00, or no limit at all. In fact, the sequence (snl can be determined in such a way that its set of limits points is an arbitrary given closed interval, bounded or not (cf. Example 2, Chapter 5). The underlying reason that this is possible is that the series of positive terms of La,. and the $eries of negative terms of La,. are both divergent I njinite Series To be specific we shall indicate a rearrangement of the series L( -l)n+i/n such that the sequence {Sn} of partial sums has the closed interval la, b) as its set of limit points. We start with the single term, and then attach negative terms: -!-!-!-! ! j until the sum first is less than a. Then we add on unused positive terms: I - 2-4" j + :3 + " k + until the Sum first is greater than b. Continuing with this idea we adjoin negative terms until the sum first is less than a ; then positive terms until the sum first exceeds b; then negative terms, etc., ad infinitum. Since the absolute value l/n of the general term (-I)n+l/n approaches zero, it follows that every number of the closed interval la, b] is approached arbitrarily closely by partial sums Sn of the rearranged series, for arbitrarily large n. Furthermore, for no number outside the interval [a, b] is this true. In the procedure just described, if the partial sums are permitted to go just above, then just below -2, then just above 3, then just below -4, etc., the sequence of partial sums of the rearranged series has the entire real number system as its set of limit points. W. Sierpinski (cf. [43]) has shown that if La,. is a conditionally convergent series with sum 8, and if s' < s, then for some rearrangement involving the positive terms only (leaving the negative terms in their original positions) the rearranged series has sum s'. A similar remark applies to numbers s" > 8 and rearrangements involving only negative terms. This is clearly an extension of the celebrated "Riemann derangement theorem" (cf. [36], p. 232, Theorem III), illustrated in all its essentials by the discussion in Example 5. In a different direction, there is an extension that reads: If La.. is a conditionally convergent series of complex numbers then the sums obtainable by all possible rearrangements that are either convergent or divergent to 00 constitute a set that is either a single line in the Complex plane (including the point at infinity) or the complex plane in toto (including the point at infinity). Furthermore, if LV.. is a 55 r : i.'...,

2 ~.. I. Fl nctiol.8 of a Real Variable c()nditionally convergent series of vectors in a finite-dimensional space, then the sums obtainable by all possible rearrangements constitute a sei; that is some linear variety in the space (cf. [47]). fl. For an arhitrary conditionally convergent series La,. and all arbitrary real number x, a sequence {En!, where I En I = for = Ii 2,..., such that Len a,. = x. Th.e procedure here is similar to that employed in Example 5., we may attach factors Cn of absolute value Since LI a,,! = + <X) in such a fashion that C a +.,. + Cn a" = I a I I a" I > x. Let r/l be the least value of n that ensures this inequality. We then p:rovide factors cn, of absolute value, for the next terms in order to obtain (for the least possible n2): el (tl C"2 a,,2 = I al I J a,. I a n +I -.,. - I a.. 2 < x. If this process is repeated, with partial sums alternately greater than x and Jess than x, a series Lcn a" is obtained which, since a" ~ 0 as n -+ + <fj must converge to x. 7. Divergent series satisfying any two of the three conditions 0 the standard alternating series theorem. The alternating series theorem referred to states that the series Lc!'., c,., where I Cn I = and en > 0, n =, 2, '",converges provided (i) En = (-)"+, n =,2,..., ( lit ", c,. + L < = en, n =, 2,.., (iii) lir:n.n_+:o en = O. No i;wo of these three conditions by themselves imply convergence; that js, no one can be omitted. The following three examples demonstrate this fact:. (i): Let c", en = lin, n =, 2,... Alternatively, for an example that is, after a fashion, an "alternating series" let {c"l be th4:l ooquence repeating in triplets:,, -,,, -,... (ii): Let c" 0= lin if n is odd, and let en 0= ljn 2 if n is even. (iii): Let en == (n + )ln (or, more simply, let en 0= ), n 2,.. 56, 6. Infinite Series 8. A divergent series whose general term approaches zero and which, with a suitable intrqduction of parentheses, becomes convergent to an arbitrary sum. Introduction of parentheses in an infinite series means grouping of consecutive finite sequences of terms (each such finite sequence consisting of at least one term) to produce a new series, whose sequence of partial sums is therefore a subsequence of the sequence of partial sums of the original series. For example, one way of introducing parentheses in the alternating harmonic series gives the series ( - ~) + G - i) +.., = (2n - ) 2n +... Any series derived from a convergent series by means of introduction of parentheses is convergent, and has a sum equal to that of the given series. The final rearranged series described under Example 5 has the stated property since, for an arbitrary real number, a suitable introduction of parentheses gives a convergent series whose sum is the given number. 9. For a given positive sequence {b,,} with limit inferior zero, a positive divergent series La,. whose general term approaches zero and such that limn-+oo an Ibn = O. Choose a subsequence bnl> b"u "', bnk' of nonconsecutive tenus of {b"l such that limk-+oo b"k = 0, and let a"k == b;k for k =, 2,... For every other value of n : n = ml, m2, m3,...,m;,..., let ami 0= Iii. Then a" ---? 0 as n ~ + <x>, La.. diverges, and a"klb"k = bn ~ 0 as k ~ + <fj. This example shows (in particular) that no matter how rapidly a positive sequence {b,,! may converge to zero, there is a positive sequence {a..l that converges to zero slowly enough to ensure divergence of the series La", and yet has a subsequence converging to zero more rapidly than the corresponding subsequence of {bnl. 0. For a given positive sequence {b n } with limit inferior zero, a positive convergent series La,. such that lim,.-+oo anlbll = + <x). 57

3 I. Functions of a Real Variable preceding, let {i,, and {jkl be strictly increas~g sequences of positive integers chosen so that the related properties hold as follows: We choose i l and il so that the related properties hold for T ; then i2 and h so that the related properties hold for both Tl and T 2 ; etc. Let the sequence {anj be defined as in Example 2. For any fixed m the sequence {ani is transformed into a sequence {b:nn } and, since the numbers i" and jk for k > m constitute sequences valid for the counterexample technique applied to T m, it. follows that limn-+co bmn does not exist for any m. 22. A power series convergent at only one point. (Cf. Example 24.) The series 2:!:'o n! x" converges for x = 0 and diverges for x 7"" O. 23. A function whose Maclaurin series converges everywhere but represents the function at only one point. The function f(x) == Ie -llx2 o if x 7"" 0, if x 0 is infinitely differentiable, all of its derivatives at x = 0 being equal to 0 (cf. Example 0, Chapter 3). Therefore its Maclaurin series +co l")(o) " +co,,_0 nl,,-0 2:--x =2:0 converges for all x to the function that is identically zero, and hence represents (converges to) the given function f only at the single point x = O. 24,. A function whose Maclaurin series converges at only one point. A function with this property is described in [0], p. 53. The function +co f(x) = 2: e- n cos n2x, n=o because of the factors e- n present in all of the series obtained by Infinite Series successive term-by-term differentiation (which therefore all converge uniformly), is an infinitely differentiable function. Its Maclaurin series has only terms of even degree, and the absolute value of the term of degree 2k is,,",xe +co 2k -J, n 40 (2)2k nx -n ~o (2k)! > 2f e for every n = 0,, 2,..., and in particular for n = 2k. For this value of n and, in terms of any given nonzero x, with k any integer greater than e/2x, we have (~;)2k e-j, = e: x )2k >. This means that for any nonzero x the Maclaurin series for f diverges. The series 2:t:o n! xn was shown in Example 22 to be convergent at only one point, x = O. It is natural to ask whether this series is the Maclaurin series for some function f(x), since an affirmative answer would provide another example of a function of the type described in the present instance. We shall now show that it is indeed possible to produce an infinitely differentiable function f(x) having the series given above as its Maclaurin series. To do this, let f",o(x) be defined as follows: For n = 2... let ", _ f (n - ))2 if 0 ~ I x I ~ 2- n /(n!)2 "o(x) =. l 0 if I x I ~ 2-n+/(n!)2 where, by means of the type of "bridging functions" constructed for Example 2, Chapter 3, no(x) is made infinitely differentiable everywhere. Let!I(x) == 0(X), and for n 2,3"", let nl(x) == lx no(t) dt, n2(x) == lx nl(t) dt, 69

4 I. l'u'rlciians of a Real Variable TlllJS jn'('j;) = CP,.,fI_2(X), f,."(x) = CP",n_3(X),.,.,f,.(n-lJ(x) = CPnO(X), f~i"')(v) = CPnO'(X). For any x and 0 ~ k ~ n - 2, i.. (k)(x) I ~ (2-+l/n:)I x,,-k- 2 /(n - k - 2)! since I CPnl(X) I ~ 2-n+ln2, j CP,,2(X) I ~ [2- n +ln 2 ] x I, The series L~:tfn(k)(X), for each Ie = 0,, 2,..., converges UJl.aarmly in every closed finite interval. Indeed, if x I ~ K, SlLd uniform convergence follows from the Weierstrass M-test (cf. [34), p. 445). Hence we see that +00 f(x) == Lf..(x) n=l is an infinitely differentiable function such that for k = 0,, 2,..., +00 focj(x) = Lfn(l,)(x). ft=l Fad ~ n ~,f..( )(0) = CPnO(H'+l)(O) = O. Forn ~ andk = n -, j,.c~)\o) = <>.. 0(0) = «n - ))2. For 0 ~ k < n -, fn(k}(o) = O. ThlS the Maclaurin series for f(x) is L!:o n! x"..2;. A convergent trigonometric series that is not a Fourier series. We shall present two examples, one in case the in+,egration involved 'is that of Riemann, and one in case the integration is that of Lebesgue. +«> The SEl'jes L SIn :X' where 0 < a ~ i, converges for every real n= n llu]tlder :x, as can be seen (d. [34], p. 533) by an application of a eootergellce test due to N. H. Abel (Norwegian, ). However, this series cannot be the Fourier series of any Riemann-integrable function f(x) since, by Bessel's inequality (cf. [34J, p. 532),?O for n =, 2,..., ~" + 2;" ~ ~ : [J(X)]2 dx. 6. Infinite Series Since f(x) is Riemann-integrable so is [j(x)]2, and the right-hand side of the preceding inequality is finite, whereas if a ~ ~ the left-hand side is unbounded as n (Contradiction.) +.,.. "smnx The senes L.,; ~l-~ also converges for every real number x. Let "=2 n n +00 f(x) == L sm nx. n=2 In n If f(x) is Lebesgue-integrable, then the function F(x) E {' f(t) dt o. is both periodic and absolutely continuous. Since f(x) is an odd function (f(x) = -f(-x)), we see that F(:r) is an even function (F(x) F( -x) and thus the Fourier series for F(x) is of the form +«> L an cos nx, n=o.. where ao = F(x) d.;, and for n ~ 2, 7r 0 an = 2.. F(x) cos nx dx 7r 0 ~ F(x) sin nx I" - 2" F' (x) sin nx dx 7r n 0 7ro n?". --- f( ) sm n:v d x x ~ --~. 7r 0 n n In n ~F'(x) exists and is equal to f(x) almost everywhere.) Since F(x) :IS of. bound~d variation, its Fourier series converges at every point, and III particular at x = 0, from which we infer that L~:2 an converges. But since an -l/(n In n), and L~:2 (-l/(n In n) diverges, we have a contradiction to the assumption that f(x) is Lebesgue-integrable. 7

5 7. Uniform Convergence Chapter 7 Uniform Convergence Intl'oduction The examples of this chapter deal with uniform convergence - and OOJlvc:ugence that is not uniform - of sequences of functions on certain sets. The basic definitions and theorems will be assumed to be known (cf. [:4], pp , [36], pp ). I. A sequence of everywhere discontinuous functions con ".n"~~ uniformly to an everywhere continuous function. f.. (x) lin if x is rational, { o if x is irrational. CLearly. lim"+-t.. f.. (x) = 0 uniformly for - 00 < x < Tills simple example serves to illustrate the following general prmeiple: Uniform convergence preserves good behavior, not bad be "f!tl-lli(ir. This same principle will be illustrated repeatedly in future 8::t;JLmples. 2. A sequence of infinitely differentiable functions converging uniformly to zero, the sequence of whose derivatives diverges eterywhel'e. Ii J", (x) == (sin nx)/yn, then since I f... (x) I ~ llyn this sequence OOJlvelges uniformly to O. To see that the sequence {in'(x)} converges nowhere, let x be fixed and consider b.. = f.. '(x) = Vii cos nx. If x == 0, b... Vii ~ + 00 as n ~ + 00, We shall show that for any x r6 0 the sequence Ibn} is unbounded, and hence diverges, by showing that there are arbitrarily large values of n such that I cos nx I ~!. Indeed, for any positive integer m such that I cos mx I <!, I cos 2mx I I 2 cos 2 mx -. I = - 2 cos 2 mx >!, SO that there exists an n > m such that I cos nx I >!. 3. A nonuniform limit of bounded functions that is not hounded. Each function f,.(x) == Illin. ( n-) 0 'x if 0 < x ~, if x = 0 is bounded on the closed interval [0, ], but the limit function f(x), equal to /x if 0 < x ~ and equal to 0 if x = 0, is unbounded there. Let it be noted that for this example to exist, the limit cannot be uniform. 4. A nonuniform lidlit of continuous functions that is not continuous. A trivial example is given by f.. (x) == {min (, nx) if x ~ 0, max (-, nx) if x < 0, whose limit is the signum function (Example 3, Chapter 3), which is discontinuous at x = O. A more interesting example is given by use of the function f (cf. Example 5, Chapter 2) defined: x =!!. in lowest terms, where p and q are integers q and q > O. x is irrational. 77

6 t. Functions of a Real Variable 7. Uniform Convergence For an arbitrary positive integer n, define f"cx) as follows: According to each pomt (~, ~), where ~ q < n, 0 ~ p ~ q, in each interval ort:h.e form (~ - 2~2' ~) define J»(x) == min (~, ~ + 2n 2 (x - ~)) j jn esc:h interval of the form (~, i + 2~2) define J,,(x) == max (~, ~ - 2n2 (x - ~)) ; and at every point x of [0, J at which J7(x) has not already been defined, let fn(x) == lin. Outside [0, J in(x) is defined so as to be periodic with period one. The graph of in(x), then, consists of an jnfinite polygonal arc made up of segments that either lie along the horiziontalline y = lin or rise with slope ±2n 2 to the isolated points of the graph of i. (Cf. Fig. 2.) As n increases, these "spikes" sharpen, and the base approaches the x axis. As a consequence, for each ::c E CR and n =, 2, &Dd lim f,,(x) = f(x), "->+00 as deuned above. Each function in is everywhere continuous, but the limit function f is discontinuous on the dense set ~ of rational numbers. (Cf. Example 24, Chapter 2.) 5. A llonuniform limit of Riemann -integrable functions that is not Riemann-integrable. (Cf. Example 33, Chapter 8.) Ea.ch function gn, defined for Example 24, Chapter 2, when restl'icted to the closed interval [0, J is Riemann-integrable there, :smee it is bounded there and has only a finite number of points of dise()ntinuity. The sequence /gn} is an increasing sequence Cgn(x) :;;:;!T.t-l(x) for each x and n, 2,... ) converging to the function f of Example, Chapter 4, that is equal to on ~ n [0, ] and equal to () on [0, ] \ Q. 7 0!!! i!! In (x) for n~5 Figure 2 6. A sequence of functions for which the limit of the integrals is not equal to the integral of the limit. Let Then but 2n 2 ;-c if 0:;;:; x ~ in (:/:) - n - 2n2( X if 2~i) 0 if -:;;:; X n lim fll(x) d:t: n-++oo 0 lim j,,(x) dx {} n~+oo 2n' < < 2n = X =, n 0 dx = O. ~ L 79

7 ~.. I. Functions of a Real Variahle Another example is the sequence {fn(x)} where fn(x) - nxe- n "" O~x~. A more extreme case is given by 2n 3 x if O~x~ 2n' 2 fn(x) - n - - < x <- ill which case, for any b E (0, ] while 2n3(X - 2~) if 2n = = n' 0 if - ~ x ~, n b lim fn(x) dx = lim?!: = + 00 n-+'" 0,,-+'" 2 ' b l lim fn(x) dx = Ib 0 dx = O. n~+~ 0 7. A sequence of functions for which the limit of the derivatives is not equal to the derivative of the limit. If fn(x) == x/(l + n2x2) for - ~ x ~ and n =,2,..., then rex) == limn-+",fn(x) exists and is equal to 0 for all x E [-,] (and this convergence is uniform since the maximum and minimum values off7l(x) on [-,] are ± /2n). The derivative of the limit is identically equal to O. However, the limit of the derivatives is I - n 2 x 2 {I if x = 0,!~fn (x) =!~'" ( + n 2 x 2 )2 = 0 if 0 < Ixl ~. 8. Convergence that is uniform on every closed subinterval but not uniform on the total interval. Let fn(x) == xn on the open interval (0, ). 9. A sequence {fn} converging uniformly to zero on [0, + 00) but such that S;"'n(X) dx +t O. 80 = lin if 0 ~ x ~ n, L e t j ( ) n X - 0 if x > n. Then f" converges uniformly to 0 on [0, + 00), but +'" fn(x) dx = ~. A more extreme case is given by f,,(x) == {~/n ~~ Then S;'" f,,(x) dx = n ~ o ~ x ~ n 2, x> n 2 7. Uniform Convergence 0. A series that converges non uniformly and whose general term approaches zero uniformly. The series 2:!: xnln on the half-open interval [0, ) has these properties. Since the general term is dominated by lin on [0, ) its uniform convergence to zero there follows immediately. The convergence of the series follows from its domination by the series 2: xn, which converges on [0, ). The nonuniformity of this convergence is a consequence of the fact that the partial sums are not uniformly bounded (the harmonic series diverges; cf. [34], p. 447, Exs. 3, 32).. A sequence converging nonuniformly and possessing a uniformly convergent subsequence. On the real number system CR, let fn(x) == { ~ if n is odd. l' f.. n 8 even. The convergence to zero is nonuniform, but the convergence of the SUbsequence {f2n(x)! = {l/2n} is uniform. 2.-Nonuniformly convergent sequences satisfying any three of the four conditions of Dini's theorem. Dini's theorem states that if {fn} is a sequence of functions defined on a set A and converging on A to a function f, and if (i) fn is continuous on A, n =, 2,..., 8