Question

determine if the series converges:

summation of n=1 to infinity of
(n + 5^n) / (n +8^n)

Answer 1

\[\sum_{n=1}^\infty \frac{n+5^n}{n+8^n}=\sum_{n=1}^\infty \frac{n}{n+8^n}+\sum_{n=1}^\infty \frac{5^n}{n+8^n}\]
I'd say comparison test for both should work.

Answer 2

direct comparison or limit comparison

Answer 3

My first instinct is direct comparison for both.

Answer 4

I think I'm doing something wrong. So I split the series into two: n/(n+8^n) and 5^n/(n+8^n) for the second I used direct comparison and i used geometric test for bn and found that the series converges. For the first, I couldn't use the direct comparison because bn>an diverted. So I used limit comparison and I got 0. But in order for it to converge it must be a finite number greater than 0. I'm stuck

Answer 5

Do ratio test. I used to do ratio test for most of my series to find convergence or divergence. It works for most series. not all, but it works for for a lot of series :P

Answer 6

I'm getting the limit = 1 which is inconclusive :/

Answer 7

are you finding the limit as the limit goes to infinity?

Answer 8

Yep

Answer 9

hmm...i have to leave now, but if no1 has helped you with this, i will come back and help you. later.

Answer 10

Ok thanks

Answer 11

@kimmy0394, what was the series you compared to for the first series?

Answer 12

Try comparing
\[\sum\frac{n}{n+8^n}\text{ to }\sum\frac{n}{8^n}\]
(You'll find that the comparison series converges by the ratio test.)

Answer 13

Oh I just did n/n . I'm confused on how to choose ur bn.

Answer 14

But that works thanks