Question

How can I tell if this series converges or not by the root test?

Answer 1

\[\sum_{k=1}^{\infty} (1+3/k)^{k^2}\]

Answer 2

Can I apply the root test twice, since if that limit converges, the other one will converge too?

Answer 3

remember that \[a _{k=1}/a _{k}\]

Answer 4

k+1****

Answer 5

Yeah, I guess for some reason I was thinking I needed to get it into a form without an exponent to see if it converged or not, but really all I have to do is evaluate the limit... right?

Answer 6

well it conv. \[\left| r \right|<1\]

Answer 7

The sum diverges. And now that I think of it, I think I was right.

Answer 8

Let me look into my Calc book.

Answer 9

That's correct, if the limit goes to infinity, the series diverges also.

Answer 10

you have to solve then take the lim if the answer is less then 1 then conv.

Answer 11

So yeah, you only need to apply it once.

Answer 12

Thanks guys, helped a bunch.

Answer 13

No problem :-)