Question

For each sequence an find a number k such that nkan
has a finite non-zero limit.
(This is of use, because by the limit comparison test the series ∑n=1∞an and ∑n=1∞n−k both converge or both diverge.)

D. a_n = ( (7n^2+7n+7)/(5n^8+3n+5√n) )^7

Answer 1

not sure just what you mean by

such that nkan has a finite non-zero limit.

Nowhere in your expression is there a place for k, and the series clearly converges, so maybe you can 'splain a bit.

Do you mean

∞
∑a_{_n-k} ??
n=1

Do you mean n^k a_{_n} ??

Answer 2

∞ ∞
∑a_n and ∑n^-k are the series that converge.
n=1 n=1

I have to find k from the a_n given, and am not sure how to do so.

Answer 3

Hmmm. I'm not sure just what you are after, either.
However, as n->∞
(7n^2+7n+7)/(5n^8+3n+5√n) -> 7n^2/5n^8 -> n^-6
because all the lower powers of n don't really matter. (nor does the pesky 7/5)
Maybe that will help some.