Abel criteria

Submitted by Structure on Sat, 11/01/2008 - 08:08.

Analysis questions

Let

$\sum_{n=0}^{n=+\infty}a_nx_n$

a series where $a_n\geq a_{n+1}..\geq 0$ ,

$\lim_{n\to \infty}a_n=0$

and $(x_n)_{n\in N}$ has partial sum $s_n=x_0+x_1+...x_n$ equal bounded. Then

$\sum_{n=0}^{n=+\infty}a_nx_n\: is\: convergent$

We shall prove the convergence using Cauchy criteria
For $\forall \epsilon>0$ there is a $n_{\epsilon}\in N$ such as $\forall n\geq n_{\epsilon}$ and $\forall p\geq 1$ we have

$|a_{n+1}x_{n+1}+a_{n+2}x_{n+2}+.....a_{n+p}x_{n+p}|\le \epsilon$

We have $x_{n+i}=s_{n+i}-s_{n+i-1}$ and as $s_n=x_0+x_1+...x_n$ are equal bounded there is a constant M such as $\forall n\:|s_n|\leq M$
We can write

$|a_{n+1}x_{n+1}+a_{n+2}x_{n+2}+.....a_{n+p}x_{n+p}|=|a_{n+1}(s_{n+1}-s_{n})+a_{n+2}(s_{n+2}-s_{n+1})+.....a_{n+p}(s_{n+p}-s_{n+p-1})|=$

$=|-a_{n+1}s_{n}+s_{n+1}(a_{n+1}-a_{n+2})+s_{n+2}(a_{n+2}-a_{n+3})+...s_{n++p-1}(a_{n+p-1}-a_{n+p})+s_{n+p}a_{n+p}|\leq 2Ma_{n+1}<\epsilon$

if n is such as

$a_{n}<\frac{\epsilon}{2M}$

Using this criteria we have the convergence of a series

$\sum_{n=1}^{n=\infty}\frac{1}{n^{\alpha}}\cos nx$

and

$\sum_{n=1}^{n=\infty}\frac{1}{n^{\alpha}} \sin nx$

for all $\alpha>0$ and $x\neq 2k\pi$

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