PlanetMath: proof of Abel's convergence theorem

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proof of Abel's convergence theorem

(Proof)

Suppose that

$\displaystyle \sum_{n=0}^\infty a_n=L$

is a convergent series, and set

$\displaystyle f(r) = \sum_{n=0}^\infty a_n r^n.$

Convergence of the first series implies that $a_n\rightarrow 0$ , and hence

converges for $\vert r\vert<1$ . We will show that $f(r)\rightarrow L$ as $r\rightarrow 1^-$ .

Let

$\displaystyle s_N=a_0+\cdots+a_N,\quad N\in\mathbb{N},$

denote the corresponding partial sums. Our proof relies on the following identity

$\displaystyle f(r)=\sum_n a_n r^n = (1-r) \sum_n s_n r^n.$

(1)

The above identity obviously works at the level of formal power series. Indeed,

$\begin{displaymath} \begin{array}{crcrcrc} &a_0 &+& (a_1+a_0)\, r &+& (a_2+a_1+a... ...s )\ = &a_0 &+& a_1\, r &+& a_2\, r^2 &+\, \cdots \end{array}\end{displaymath}$

Since the partial sums

converge to

, they are bounded, and hence $\sum_n s_n r^n$ converges for $\vert r\vert<1$ . Hence for $\vert r\vert<1$ , identity (1) is also a genuine functional equality.

Let $\epsilon>0$ be given. Choose an sufficiently large so that all partial sums, with , satisfy $\vert s_n-L\vert\le\epsilon$ . Then, for all such that , one obtains

$\displaystyle \left\vert\sum_{n=N+1}^\infty (s_n - L) r^n\right\vert \le \epsilon\,\frac{r^{N+1}}{1-r}\,.$

Note that

$\displaystyle f(r) - L = (1-r)\sum_{n=0}^N (s_n - L) r^n + (1-r) \sum_{n=N+1}^\infty (s_n - L) r^n.$

As $r\rightarrow 1^-$ , the first term tends to 0. The absolute value of the second term is estimated by $\epsilon\, r^{N+1}\le \epsilon$ . Hence,

$\displaystyle \limsup_{r\rightarrow 1^-} \vert f(r) - L\vert \le \epsilon.$

Since $\epsilon>0$ was arbitrary, it follows that $f(r)\rightarrow L$ as $r\rightarrow 1^-$ . QED

"proof of Abel's convergence theorem" is owned by rmilson. [ full author list (2) ]

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See Also: proof of Abel's limit theorem

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Cross-references: QED, absolute value, functional equality, bounded, formal power series, level, identity, proof, partial sums, converges, implies, series, convergent series

This is version 6 of proof of Abel's convergence theorem, born on 2002-11-02, modified 2008-08-07.
Object id is 3562, canonical name is ProofOfAbelsConvergenceTheorem.
Accessed 4999 times total.

Classification:

AMS MSC:

40G10 (Sequences, series, summability :: Special methods of summability :: Abel, Borel and power series methods)

Pending Errata and Addenda

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