A COURSE OF MODERN ANALYSIS

8.4 Methods of ‘summing’ series.

We have seen that it is possible to obtain a development of the form \[ f(x) = \sum_{m=0}^{n} A_mx^{-m} + R_n(x), \] where \(R_n (x) \rightarrow \infty\) as \(n \rightarrow\infty\), and the series \(\sum\limits_{m=0}^{\infty} A_m x^{-m}\) does not converge.

We now consider what meaning, if any, can be attached to the ‘sum’ of a non-convergent series. That is to say, given the numbers \(a_0\), \(a_1\), \(a_2, \dots\), we wish to formulate definite rules by which we can obtain from them a number \(S\) such that \(S = \sum_{n=0}^{\infty} a_n\) if \(\sum_{n=0}^{\infty} a_n\) converges, and such that \(S\) exists when this series does not converge.

8.41 Borel’s method of summation.[1]

We have seen (§7.81) that \[ \sum_{n=0}^{\infty} a_n z^n = \! \int_0^{\infty} \! e^{-t} \phi(tz) \, d t , \] where \(\phi(tz) = \sum\limits_{n=0}^{\infty} \dfrac{a_n t^nz^n}{n!} \), the equation certainly being true inside the circle of convergence of \(\sum\limits_{n=0}^{\infty} a_n z^n\) . If the integral exists at points \(z\) outside this circle, we define the ‘Borel sum’ of \(\sum\limits_{n=0}^{\infty} a_n z^n\) to mean the integral.

Thus, whenever \(\mathfrak{Re}(z) < 1\), the ‘Borel sum’ of the series \(\sum\limits_{n=0}^{\infty} z^n\) is \[ \int_0^\infty \! e^{-t}e^{tz} \, d t = (1-z)^{-1}. \] If the ‘Borel sum’ exists we say that the series is ‘summable \((B)\)’.

8.42 Euler’s method of summation.[2]

A method, practically due to Euler, is suggested by the theorem of §3.71; the ‘sum’ of \(\sum\limits_{n=0}^{\infty} a_n\) may be defined as \(\lim\limits_{x \rightarrow 1-0}\;\! \sum\limits_{n=0}^{\infty} a_n x^n\), when this limit exists.

Thus the ‘sum’ of the series \(1 - 1 + 1 - 1 + \cdots\) would be \[ \lim_{x \rightarrow 1-0} (1 - x + x^2 - \cdots ) = \lim_{x \rightarrow 1-0} (1 + x)^{-1} = \frac{1}{2}. \]

8.43 Cesàro’s method of summation.[3]

Let \(s_n = a_1 + a_2 + \cdots + a_n\); then if \(S =\lim\limits_{n \rightarrow \infty} \dfrac{1}{n}(s_1 + s_2 + \cdots + s_n )\) exists, we say that \(\sum_{n=1}^{\infty} a_n\) is ‘summable \((C\:\! 1)\)’, and that its sum \((C\:\! 1)\) is \(S\). It is necessary to establish the ‘condition of consistency’,[4] namely that \(S = \sum_{n=1}^{\infty} a_n\) when this series is convergent .

To obtain the required result, let \(\sum\limits_{m=1}^{\infty} a_m = s\), \(\sum\limits_{m=1}^{\infty} s_m = nS_{n\:\!\!}\); then we have to prove that \(S_n \rightarrow s\),

Given \(\epsilon\), we can choose \(n\) such that \(\left| \, \sum\limits_{m=m+1}^{n+p} a_m \, \right| < \epsilon \) for all values of \(p\), and so that \(\left| \, s - s_n \, \right| < \epsilon \:\!\!\).

Then, if \(\nu > n\), we have \[ \begin{align*} S_{\nu}= a_1 &+ a_2 \left(1 - \frac{1}{\nu} \right) + \cdots + a_n \left(1 - \frac{n-1}{\nu} \right) \\ &+ a_{n+1} \left(1 - \frac{n}{\nu} \right)+ \cdots + a_{\nu} \left(1 - \frac{\nu-1}{\nu} \right) . \end{align*} \] Since \(1\), \(1 - v^{-1}\), \(1 - 2v^{-1}, \dots\) is a positive decreasing sequence, it follows from Abel’s inequality (§2.301) that \[ \left| \, \:\! a_{n+1} \left(1 - \frac{n}{\nu} \right) + a_{n+2} \left(1 - \frac{n+1}{\nu} \right) + \cdots + a_{\nu} \left(1 - \frac{\nu-1}{\nu} \right) \, \right| < \left( 1 - \frac{n}{\nu} \right) \epsilon . \] Therefore \[ \left| \, \:\! S_{\nu} - \left\{ a_1 + a_2 \left(1 - \frac{1}{\nu} \right) + \cdots + a_n \left(1 - \frac{n-1}{\nu} \right) \right\} \, \right | < \left( 1 - \frac{n}{\nu} \right) \epsilon . \]

Making \(\nu \rightarrow\infty \), we see that, if \(S\) be any one of the limit points (§2.21) of \(S_{\nu}\), then \[ \left| \, S- \sum_{m=1}^n a_m \, \right| \leq \epsilon . \] Therefore, since \(\left|\, s - s_n \,\right| \leq \epsilon \;\!\! \), we have \[ \left| \, S-s \, \right| \leq 2 \epsilon . \] This inequality being true for every positive value of \(\epsilon\) we infer, as in §2.21, that \(S = s\); that is to say \(S_{\nu}\) has the unique limit \(s\); this is the theorem which had to be proved.

Example 1. Frame a definition of ‘uniform summability \((C\:\! 1)\) of a series of variable terms.’

Example 2. If \(b_{n,\;\!\nu} \geq b_{n+1,\;\!\nu} \geq 0\) when \(n< \nu\) and if, when \(n\) is fixed, \(\lim\limits_{\nu \rightarrow \infty} b_{n,\;\!\nu} = 1\), and if \(\sum\limits_{m=1}^{\infty} a_m =s\), then \( \displaystyle \lim_{\nu \rightarrow \infty} \left\{ \sum_{n=1}^{\nu} a_n b_{n,\;\!\nu} \right\} =s.\)

8.431 Cesàro’s general method of summation.

A series \(\sum\limits_{n=1}^{\infty} a_n\) is said to be ‘summable \((C\:\! r)\)’ if \(\lim\limits_{\nu \rightarrow \infty} \sum\limits_{n=1}^{\nu} a_n b_{n,\;\!\nu}\) exists, where \[ \begin{align*} b_{0,\;\!\nu} &=1 \\ b_{n,\;\!\nu} &= \left\{ \left( 1 + \frac{r}{v+1-n} \right) \left( 1 + \frac{r}{v+2-n} \right) \cdots \left( 1 + \frac{r}{v-1} \right) \right\}^{-1} \! . \end{align*} \] It follows from §8.43 example 2 that the ‘condition of consistency’ is satisfied; in fact it can be proved[5] that if a series is summable \((C\:\! r')\) it is also summable \((C\:\! r)\) when \(r > r' \!\); the condition of consistency is the particular case of this result when r = 0.

8.44 The method of summation of Riesz.[6]

A more extended method of ‘summing’ a series than the preceding is by means of \[ \lim_{\nu \rightarrow \infty} \;\! \sum_{n=1}^\infty \left( 1- \frac{\lambda_n}{\lambda_\nu} \right)^{\:\!\! r} a_n . \] in which \(\lambda_n\) is any real function of \(n\) which tends to infinity with \(n\). A series for which this limit exists is said to be ‘summable \((R\:\! r)\) with sum-function \(\lambda_n\)’.

8.5 Hardy’s convergence theorem.[7]

Let \(\sum\limits_{n=1}^\infty a_n \) be a series which is summable \((C\:\! 1)\). Then if \[ a_n = O\left( 1 \middle/ n \right), \] the series \(\sum\limits_{n=1}^\infty a_n \) converges.

Let \(s_n = a_1 + a_2 + \cdots + a_n\) then since \(\sum\limits_{n=1}^\infty a_n \) is summable \((C\:\! 1)\), we have \[ s_1 + s_2 + \cdots + s_n = n\left\{ s + o(1) \right\}, \] where \(s\) is the sum \((C\:\! 1)\) of \(\sum\limits_{n=1}^\infty a_n \).

Let \[ s_m - s = t_m, \quad (\, m = 1, \, 2, \cdots , \:\! n\,), \] and let \[ t_1 + t_2 + \cdots + t_n = \sigma_n . \]

With this notation, it is sufficient to shew that, if \(\left|\, a_n \,\right| < Kn^{-1}\), where \(K\) is independent of \(n\), and if \(\sigma_ n = n\cdot o (1)\), then \(t_n \rightarrow 0\) as \(n \rightarrow \infty\).

Suppose first that \(a_1\), \(a_2, \dots\) are real. Then, if \(t_n\) does not tend to zero, there is some positive number \(h\) such that there are an unlimited number of the numbers \(t_n\) which satisfy either (i) \(t_n > h\) or (ii) \(t_n < -h\). We shall shew that either of these hypotheses implies a contradiction. Take the former,[8] and choose \(n\) so that \(t_n > h\).

Then, when \(r = 0\), \(1\), \(2\), \(\dots\), \[ \left|\, a_{n+1} \,\right| < \left. \vphantom{z} K \middle/ n \right. . \] Now plot the points \(P_r\) whose coordinates are \((r, t_{\:\! n+r})\) in a Cartesian diagram. Since \(t_{\:\! n+r+1}-t_{\:\! n+r} = a_{n+r+1}\), the slope of the line \(P_r P_{r+1}\) is greater than than \(-\left. K \middle/ n \right.\). Let \(\theta= \arctan (\left. K \middle/ n \right.)\).

Therefore the points \(P_0\), \(P_1\), \(P_2, \dots\) lie above the line \(y = h - x \tan\theta\). Let \(P_k\) be the last of the points \(P_0\), \(P_1, \dots\), which lie on the left of \(x=h \cot\theta\), so that \(k \leq h \cot\theta\).

Draw rectangles as shewn in the figure. The area of these rectangles exceeds the area of the triangle bounded by \(y = h - x \tan \theta\) and the axes; that is to say \[\begin{align*} \sigma_{n+k} - \sigma_{n-1} &= t_n + t_{n+1} + \cdots + t_{n+k} \\ &> \frac{1}{2} h^2 \cot \theta = \frac{1}{2} h^2 K^{-1} n. \end{align*}\]

But \[\begin{align*} \left|\, \sigma_{n+k} - \sigma_{n-1} \,\right| & \leq \left|\, \sigma_{n+k} \,\right| + \left|\, \sigma_{n-1} \,\right| \\ &= (n + k)\cdot o(1) + (n-1)\cdot o(1) \\ &= n\cdot o(1), \end{align*}\] since \(k \leq hnK^{-1}\), and \(h\), \(K\) are independent of \(n\).

Therefore, for a set of values of \(n\) tending to infinity, \[ \frac{1}{2} h^2 K^{-1} < n \cdot o(1), \] which is impossible since \(\frac{1}{2} h^2 K^{-1}\) is not \(o(1)\) as \(n \rightarrow \infty\).

This is the contradiction obtained on the hypothesis that \(\varlimsup t_n \geq h > 0\); therefore \(\varlimsup t_n \leq 0\). Similarly, by taking the corresponding case in which \(t_n \leq - h\), we arrive at the result \(\varliminf t_n \geq 0\). Therefore since \(\varlimsup t_n \geq \varliminf t_n \), we have \[ \varlimsup t_n = \varliminf t_n = 0, \] and so \[ t_n \rightarrow 0. \] That is to say \(s_n \rightarrow s\), and so \(\sum\limits_{n=1}^\infty a_n \) is convergent and its sum is \(s\).

If \(a_n\) be complex, we consider \(\mathfrak{Re} (a_n)\) and \(\mathfrak{Im} (a_n)\) separately, and find that \(\sum\limits_{n=1}^\infty \mathfrak{Re} (a_n) \) and \(\sum\limits_{n=1}^\infty \mathfrak{Im} (a_n) \) converge by the theorem just proved, and so \(\sum\limits_{n=1}^\infty a_n \) converges.

The reader will see in Chapter ix that this result is of great importance in the modern theory of Fourier series.

Corollary. If \(a_{n}(\xi)\) be a function of \(\xi\) such that \(\sum\limits_{n=1}^\infty a_{n}(\xi) \) is uniformly summahle \((C\:\! 1)\) throughout a domain of values of \(\xi\), and if \(\left|\, a_{n}(\xi) \,\right| < Kn^{-1}\), where \(K\) is independent of \(\xi\), then \(\sum\limits_{n=1}^\infty a_{n}(\xi) \) converges uniformly throughout the domain.

For, retaining the notation of the preceding section, if \(t_{n}(\xi)\) does not tend to zero uniformly, we can find a positive number \(h\) independent of \(n\) and \(\xi\) such that an infinite sequence of values of \(n\) can be found for which \(t_{n}(\xi_{n}) > h\) or \(t_{n}(\xi_{n}) > -h\) for some point \(\xi_{n}\) of the domain;[9] the value of \(\xi_n\) depends on the value of \(n\) under consideration.

We then find, as in the original theorem, \[  \frac{1}{2} h^2 K^{-1} < n \cdot o(1), \] for a set of values of \(n\) tending to infinity. The contradiction implied in the inequality shews that \(h\) does not exist,[10] and so \(t_{n}(\xi) \rightarrow 0\) uniformly.

References.

H. Poincaré Acta Mathematica, viii. (1886), pp. 295–344.

E. Borel, Leçons sur les Séries Divergentes (Paris, 1901).

T. J. I’a. Bromwich, Theory of Infinite Series (1908), Ch. xi.

E. W. Barnes, Phil. Trans. of the Royal Society, 206, a (1906), pp. 249–297.

G. H. Hardy and J. E. Littlewood, Proc. London Math. Soc. (2), xi. (1913), pp. 1–16.[11]

G. N. Watson, Phil. Trans. of the Royal Society, 211, a (1912), pp. 279–313.

S. Chapman, Proc. London Math. Soc. (2), ix. (1911), pp. 369–409.[12]