Question

How can I tell if this series converges or diverges?

Answer 1

\[\sum_{1}^{\infty} \cos(n pi)/(n^.75)\]

Answer 2

Well if you go to infinity will the value go to a constant value? if it does it diverges. If the value goes to inifnity as x goes to inifnity then it converges.

Answer 3

Its a fraction.. still cant get it to show up right. But yeah how do I do it since the top varies from -1 to 1? I thought comparison, but I guess not?

Answer 4

Does this make sense to anyone? Would I see alternative series?Or does that still not work

Answer 5

use a/1-r - a being the first term and r being the ratio, which is the number the preceding term is multiplied by. If the answer is less than 1 and greater than -1 it is converging.

Answer 6

Gaah, OS crashed on me. Notice that this series is smaller than 1/n^75. Remember that if we have a series {an} <= {bn} and bn converges, then {an} also converges. Prove that 1/n^75 converges and voila! you are done.

Answer 7

If you prefer, take the absolute value of the cos, it'll still be smaller than 1/n^75. If a series absolutely converges, then it converges.

Answer 8

To be clear:\[\frac{|cosn|}{n^{75}} \le \frac{1}{n^{75}}\]because |cosn| will vary between [0,1] while the RHS is always 1.