3.7 Uniformity of convergence of power series.
Let the power series \[a_0+a_1z+ \cdots +a_nz^n+ \cdots\] converge absolutely when \(z = z_0\). Then, if \(\left|\,z \,\right| \leq \left|\, z_0 \,\right|\), \[\left|\, a_n z^n \,\right| \leq \left|\, a_nz_0^n \,\right|.\] But since \(\sum\limits_{n=0}^{\infty} \left|\,a_nz_0^n \,\right|\) converges, it follows, by §3.34, that \(\sum\limits_{n=0}^{\infty} \left|\, a_nz^n \,\right|\) converges uniformly with regard to the variable \(z\) when \(\left|\,z \,\right| \leq \left|\,z_0 \, \right|\).
Hence, by §3.32, a power series is a continuous function of the variable throughout the closed region formed by the interior and boundary of any circle concentric with the circle of convergence and of smaller radius (§2.6).
3.71 Abel’s theorem[1] on continuity up to the circle of convergence.
Let \(\sum\limits_{n=0}^{\infty} a_n z^n\) be a power series, whose radius of convergence is unity, and let it be such that \(\sum\limits_{n=0}^{\infty} a_n \) converges; and let \(0 \leq x \leq 1\); then Abel’s theorem asserts that \(\lim\limits_{x \rightarrow 1-0} \left(\sum\limits_{n=0}^{\infty} a_n x^n \right)=\sum\limits_{n=0}^{\infty} a_n \).
For, with the notation of §3.35, the function \(x^n\) satisfies the conditions laid on \(u_n (x)\), when \(0 \leq x \leq 1\); consequently \(f(x) = \sum\limits_{n=0}^{\infty} a_n x^n\) converges uniformly throughout the range \(0 \leq x \leq 1\); it is therefore, by §3.32, a continuous function of \(x\) throughout the range, and so \(\lim\limits_{x \, \rightarrow 1-0} f(x) =f(1)\), which is the theorem stated.
3.72 Abel’s theorem[2] on multiplication of series.
This is a modification of the theorem of §2.53 for absolutely convergent series.
Let \[c_n = a_0b_n + a_1b_{n-1}+ \cdots + a_nb_0.\] Then the convergence of \(\sum\limits_{n=0}^{\infty} a_n,\: \sum\limits_{n=0}^{\infty} b_n \) and \(\sum\limits_{n=0}^{\infty} c_n\) is a sufficient condition that \[\left(\sum\limits_{n=0}^{\infty} a_n \right) \left(\sum\limits_{n=0}^{\infty} b_n \right) = \sum\limits_{n=0}^{\infty} c_n.\]
For, let \[A(x)=\sum\limits_{n=0}^{\infty} a_n x^n, \quad B(x)=\sum\limits_{n=0}^{\infty} b_n x^n, \quad C(x)=\sum\limits_{n=0}^{\infty} c_n x^n.\] Then the series for \(A (x),\: B(x),\: C(x)\) are absolutely convergent when \(\left|\, x\, \right| < 1\), (§2.6); and consequently, by §2.53, \[A(x)\, B(x)=C(x)\] when \(0 < x < 1\); therefore, by §2.2 example 2, \[\left\{ \lim_{x \rightarrow 1-0} A(x)\right\}\left\{ \lim_{x \rightarrow 1-0} B(x)\right\} = \left\{ \lim_{x \rightarrow 1-0} C(x)\right\} \] provided that these three limits exist; but, by §3.71, these three limits are \(\sum\limits_{n=0}^{\infty} a_n,\: \sum\limits_{n=0}^{\infty} b_n, \:\sum\limits_{n=0}^{\infty} c_n\); and the theorem is proved.
3.73 Power series which vanish identically.
If a convergent power series vanishes for all values of \(z\) such that \(\left|\, z\,\right| < r_{\:\! 1}\), where \(r_{\:\! 1} > 0\), then all the coefficients in the power series vanish.
For, if not, let \(a_m\) be the first coefficient which does not vanish.
Then \(a_m +a_{m+1} z+a_{m+2} z^2 + \cdots\) vanishes for all values of \(z\) (zero excepted) and converges absolutely when \(\left| \,z\, \right| \leq r < r_1\) (§2.6); hence, if \(s = a_{m+1} + a_{m+2} z + \cdots \), we have \[\left|\, s \, \right| \leq \sum_{n=1}^{\infty} \,\left|\, a_{m+n} \right|\, r^{n-1},\] and so we can find[3] a positive number \( \delta \leq r \) such that, whenever \(\left| \, z \,\right| \leq \delta \), \[\left|\,s z \,\right|=\left|\, a_{m+1} z+a_{m+2} z^2 + \cdots \,\right| \leq \frac{1}{2}\left|\,a_m \right|;\] and then \(\left|\,a_m + sz\,\right|\geq \left|\, a_m \right|- \left|\,sz\,\right| \geq \frac{1}{2} \left|\, a_m \right|\), and so \(\left|\,a_m + sz\,\right| \neq 0 \) when \(\left| \, z \,\right| \leq \delta\). We have therefore arrived at a contradiction by supposing that some coefficient does not vanish. Therefore all the coefficients vanish.
Corollary 1. We may ‘equate corresponding coefficients’ in two power series whose sums are equal throughout the region \(\left|\,z\, \right|< \delta\), where \(\delta > 0\).
Corollary 2. We may also equate coefficients in two power series which are proved equal only when \(z\) is real.
References.
- Complex functions and uniformity.
- T. J. l’a. Bromwich, Theory of Infinite Series (1908), Ch. vii.
- E. Goursat, Cours d’Analyse (Paris, 1910, 1911) Chs. i, xiv.
- C. J. de la Vallée Poussin, Cours d’Analyse Infinitésimale (Louvain and Paris, 1914), Introduction and Ch. viii.
- G. H. Hardy, A course of Pure Mathematics (1914), Ch. v.
- W. F. Osgood, Lehrbuch der Funktionentheorie (Leipzig, 1912), Chs. ii, iii.
- G. N. Watson, Complex Integration and Cauchy’s Theorem (Camb. Math. Tracts, No. 15), (1914), Chs. i, ii.
Miscellaneous Examples.
Shew that the series \[\sum_{n=1}^{\infty}\frac{z^{n-1}}{(1-z^n)(1-z^{n+1})}\] is equal to \(\displaystyle \frac{1}{(1-z)^2}\) when \( \left|\, z\, \right| < 1\) and is equal to \(\displaystyle \frac{1}{z(1-z)^2}\) when \(\left|\, z\, \right| > 1\). Is this fact connected with the theory of uniform convergence ?
Shew that the series \[2\sin \frac{1}{3z} + 4 \sin \frac{1}{9z} + \cdots + 2^n\sin \frac{1}{3^n z} + \cdots \] converges absolutely for all values of \(z\) (\(z = 0\) excepted), but does not converge uniformly near \(z = 0\).
If \[u_n(x)=-2(n-1)^2xe^{-(n-1)^2 x^2} + 2n^2xe^{-n^2x^2},\] shew that \(\sum\limits_{n=1}^{\infty} u_n(x)\) does not converge uniformly near \(x=0\). \(\vphantom{\\ 3\\}\)
(Math. Trip., 1907.)Shew that the series \(\displaystyle \frac{1}{\sqrt{1}} - \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} - \cdots\) is convergent, but that its square (formed by Abel’s rule) \[\frac{1}{1} - \frac{2}{\sqrt{2}} +\left(\frac{2}{\sqrt{3}} + \frac{1}{2} \right) - \left(\frac{2}{\sqrt{4}} + \frac{2}{\sqrt{6}} \right) + \cdots\] is divergent.
If the convergent series \(\displaystyle s=\frac{1}{1^r}-\frac{1}{2^r}+\frac{1}{3^r}-\frac{1}{4^r}+ \cdots \;\;(r>0)\:\!\) be multiplied by itself, the terms of the product being arranged as in Abel’s result, shew that the resulting series diverges if \(r \leq \frac{1}{2}\) but converges to the sum \(s^2\) if \(r > \frac{1}{2}\). \(\vphantom{\\ 3\\}\)
(Cauchy and Cajori.)If the two conditionally convergent series \[\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^r}\,\text{ and }\, \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^s},\] where \(r\) and \(s\) lie between 0 and 1, be multiplied together, and the product arranged as in Abel’s result, shew that the necessary and sufficient condition for the convergence of the resulting series is \(r + s > 1\). \(\vphantom{\\ 3\\}\)
(Cajori.)Shew that if the series \(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7} + \cdots \) be multiplied by itself any number of times, the terms of the product being arranged as in Abel’s result, the resulting series converges. \(\vphantom{\\ 3\\}\)
(Cajori.)Shew that the \(q\)th power of the series \[a_1\sin \theta+a_2 \sin 2\theta + \cdots + a_n \sin n\theta + \cdots\] is convergent whenever \(q (1 - r)< 1\), \(r\) being the greatest number satisfying the relation \[a_n \leq n^{-r}\] for all values of \(n\).
Shew that if \(\theta\) is not equal to 0 or a multiple of \(2\pi\), and if \(u_0,\, u_1,\, u_2,\, \dots \) be a sequence such that \(u_n \rightarrow 0\) steadily, then the series \(\sum u_n \cos(n\theta + \phi) \) is convergent.
Shew also that, if the limit of \(u_n\) is not zero, but \(u_n\) is still monotonic, the sum of the series is oscillatory if \(\left.\theta \middle/\pi \right.\) is rational, but that, if \(\left.\theta \middle/\pi \right.\) is irrational, the sum may have any value between certain bounds whose difference is \(a \,\mathrm{cosec} \frac{1}{2}\theta\), where \(a= \lim\limits_{n \rightarrow \infty} u_n\). \(\vphantom{\\ 3\\}\)
(Math. Trip., 1896.)