# alternating series sum

• Nov 23rd 2012, 10:55 PM
prasum
alternating series sum
calculate the sum of (-1)^n (n/8^n)

i know fourier coefficients and bessels inequality how to evaluate this
• Nov 24th 2012, 03:38 AM
sbhatnagar
Re: alternating series sum
The sum, $\displaystyle \sum_{n=1}^{\infty} (-1)^n\frac{n}{8^n}$, can be evaluated by using only elementary techniques.

Let $\displaystyle S=\sum_{n=1}^{\infty} (-1)^n\frac{n}{8^n}$...(1)

$\displaystyle 8S=\sum_{n=1}^{\infty} (-1)^n\frac{n}{8^{n-1}}$...(2)

Adding (1) and (2) gives the incredible result

$\displaystyle 9S=-1+\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{8^n}=-1+\frac{1}{9}$
$\displaystyle =\frac{-8}{9}$

so $\displaystyle S=\frac{-8}{81}$
• Nov 24th 2012, 04:02 AM
prasum
Re: alternating series sum
i did not understand the step which produces the incredible result can yu please explain
• Nov 24th 2012, 04:45 AM
sbhatnagar
Re: alternating series sum
Quote:

Originally Posted by prasum
i did not understand the step which produces the incredible result can yu please explain

$\displaystyle 8S=-1+\frac{2}{8}-\frac{3}{8^2}+\cdots$

$\displaystyle S=-\frac{1}{8}+\frac{2}{8^2}-\frac{3}{8^3}+\cdots$
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$\displaystyle 9S=-1+\frac{1}{8}-\frac{1}{8^2}+\frac{1}{8^3}-\cdots$

$\displaystyle =-1+\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{8^n}$
• Nov 24th 2012, 04:49 AM
Plato
Re: alternating series sum
Quote:

Originally Posted by prasum
i did not understand the step which produces the incredible result can yu please explain

You posted this in the calculus forum. So I assume it should be done using calculus.

If $\displaystyle y = \sum\limits_{k = 1}^\infty {r^k } = \frac{r}{{1 - r}},\;\left| r \right| < 1$ then $\displaystyle y' = \sum\limits_{k = 1}^\infty {kr^{k - 1} } = \frac{1}{{\left( {1 - r} \right)^2 }}$.

The multiply by $\displaystyle r$ to get $\displaystyle \sum\limits_{k = 1}^\infty {kr^{k} } = \frac{r}{{\left( {1 - r} \right)^2 }}$.

In this question $\displaystyle r=\frac{-1}{8}$.