- #1
MillerGenuine
- 64
- 0
Homework Statement
[tex]
\sum_{k=2}^\infty \frac{k^2}{k^2-1}
[/tex]
[tex]
\sum_{n=1}^\infty \frac{1+2^n}{3^n}
[/tex]
Homework Equations
I know that for the first problem i can apply the Divergence test by finding my limit as K goes to infinity. By doing this i get 1 which does not equal zero so i know it diverges.
Now my question is why can't i apply this same test to the 2nd problem? it seems as n approaches infinity i would get 1 as well. but that's not the case the correct solution for the 2nd problem is as follows..
[tex]
\sum_{n=1}^\infty \frac{1}{3^n} + \frac{2^n}{3^n}
[/tex]
Then by using a little algebra and sum of two convergent geometric series we get 5/2 to be the answer. which is convergent.
So why can't i find my limit as n approaches infinity in the 2nd problem? and why is 5/2 convergent? i thought if r>1 the series diverges?
Any help is appreciated.