Question

Does the series converge or diverge?

n/(n^2+1) n=1 to infinity

Studying for an exam I just need an answer on this one please. Professor decided theres no need for an answer key for our study sheet....idiot.

Answer 1

Diverge. Since:
\[
\frac{n}{n^2+1}\sim\frac{1}{n}
\]And:
\[
\sum_n\frac{1}{n}\to \infty
\]

Answer 2

i used the limit comparison test and got 1 so it diverges because of the p-series that states p<or equal to 1 diverges. can i do it that way too?

Answer 3

You can't quite use limit comparison since:
\[
\frac{1}{n}>\frac{n}{n^2+1}=\frac{1}{n+\frac{1}{n}}
\]

Answer 4

i think i did it wrong then haha i picked the wrong Bn

thank you:)

Answer 5

For this, you should probably make use of the ratio test.

And, yup.

Answer 6

when can i NOT use the ratio test?

Answer 7

Generally, some cases are fairly complicated, and, if you have a better way of doing it, then that's fine---also it's possible for the ratio test to be inconclusive, if, for example, you have:
\[
\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=1
\]
Otherwise, you should probably make use of the ratio test.

Unless, of course, you want to find some nice way of doing the problem, which there are always many.

Answer 8

awesome thank you:)

Answer 9

Yup.