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[parent] 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 $ f(r)$ 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 $ s_n$ converge to $ L$, 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 $ N$ sufficiently large so that all partial sums, $ s_n$ with $ n>N$, satisfy $ \vert s_n-L\vert\le\epsilon$. Then, for all $ r$ such that $ 0<r<1$, 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 MSC40G10 (Sequences, series, summability :: Special methods of summability :: Abel, Borel and power series methods)

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