# Thread: alternating series sum

1. ## alternating series sum

calculate the sum of (-1)^n (n/8^n)

i know fourier coefficients and bessels inequality how to evaluate this

2. ## 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 $\displaystyle S=\frac{-8}{81}$

3. ## Re: alternating series sum

i did not understand the step which produces the incredible result can yu please explain

4. ## Re: alternating series sum

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

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

5. ## Re: alternating series sum

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