Sum of an Infinite Series

[waealu]

Homework Statement

(Sorry, but I haven't mastered using the sigma notation in these forums yet).

Find the sum of the following infinite series: (n=0)^(inf) SIGMA ((pi)cos(n))/(5^n).

Homework Equations

I tried using the formula S=(a₁)/(1-r).

I know that a=pi, but I can't find "r." The equation doesn't have a consistent rate. Should I be using a different method?

Thanks.

 

[micromass]

Hi waealu!

The sum you have wrote down is not a geometric series, so the formula you mentioned is not applicable (yet).

The problem is the cos(n), that spoils the fun. But there is a way to change the cos(n) into exponents by using the formula

$$\cos(n)=\frac{(e^i)^n+(e^{-i})^n}{2}$$

With this equation, you can express your series as a (sum of) geometric series.

Note, to display the sum with LaTeX, type

[ tex ] \sum_{n=0}^{+\infty} \frac{\pi \cos(n)}{5^n} [ /tex ]

without the spaces in the [ tex ] tags.