one more series (finding convergence)

[iDoof]

i'm looking at this series:

SUM of (n/(n+1))^(n^2) from n=1 to infinity

what test should i use to test its convergence? i was thinking Root test, but i'm thrown off by the n^2....is there a way to simplify it that i don't see?


thanks again to all for your time!

 

[galactus]

You can use the ratio test for this one too.

\(\displaystyle (\frac{n+1}{n})^{n^{2}}(\frac{n+1}{n+2})^{(n+1)}^{2}\)

If you take this limit, you will find it converges. But to what?.

 

[pka]

This is a great one for the ROOT TEST.

\(\displaystyle \L
\sqrt[n]{{\left( {\frac{n}{{n + 1}}} \right)^{n^2 } }} = \left( {\frac{n}{{n + 1}}} \right)^n \to \frac{1}{e}\)

 

[galactus]

That's what I got using the ratio test. Cool. I ran the sum through Maple to see what it actually converged to. Pretty cool. It converges to \(\displaystyle \frac{80609}{98614}\)