Does this infinite series converge or diverge?

cos(pi) + cos(2pi)/2 + cos(3pi)/3 + cos(4pi)/4 + .....

= -1 + 1/2 -1/3 + 1/4 - 1/5 + 1/6 ...

equals -ln(2)

http://en.wikipedia.org/wiki/Natural_logarithm_of_...

Cute.

It converges. The decrease as n-> infinity will be 0. you are able to ensure this out with the help of searching on the powers contained in the numerator and denominator. The numerator will strengthen at a slower fee than the denominator because the n contained in the denominator is raised to the third means and the numerator in raised to the second one means. ex. n/(n^2) will converge to 0. (n^2)/n will diverge if the powers are an similar something a touch distinct takes position: 2(n^2)/(n^2) converges to 2