How do you find out where a series converges

[stunner5000pt]

are there certain formulae to find out where a certain infinite series converges, if it does converge

for example
\sum_{n=1}^\infty (\frac{3}{5})^n certainly converges because it is between the infinite series
\sum (1+ \frac{1}{n})^n and the series \sum (\frac{1}{5})^n which both converge Since both of them converge then sum(3/5)^n must converge.

But my question is WHERE does (3/5)^n converge??

 

[Hurkyl]

\sum_{n=1}^\infty (3/5)^n doesn't have any parameters, so it doesn't really make sense to ask for which values of the parameters (i.e. where) the series converges... only if.


Now, (1 + 1/n)^n and (1/5)^n aren't series...

 

[stunner5000pt]

what i meant was how do you find the explicit value of the series
for example i know (because I've been told) that
\sum \frac{1}{n^2} = \frac{\pi^2}{6}
would it be possible to that to do this series??

note: my proof of its convergence is wrong

 

[Hurkyl]

\sum_{n = 1}^{\infty} (1 + 1/n)^n doesn't converge...

Anyways, yes, your series does have a sum. It's a geometric series, so use the formula for such series.

 

[saltydog]

what i meant was how do you find the explicit value of the series
for example i know (because I've been told) that
\sum \frac{1}{n^2} = \frac{\pi^2}{6}
would it be possible to that to do this series??

note: my proof of its convergence is wrong

Here's the thread that addressed the problem:
https://www.physicsforums.com/showthread.php?p=470773#post470773