Testing Series for Convergence

[rheighton]

Hi everybody,
I'm really stuck on some problems testing the convergence of these sequences:

1) (n>=0)∑ [(n!) / (e^(n))]

2) (n>=0)∑ [(2^(n)) / (1+e^(n)]

3) (n>=1)∑ [(nln(n)) / (1+n^(3))]

Help would really be appreciated!

 

[Daniel_Feldman]

So each one of these goes to infinity? I mean, these are series right, not sequences.


If so,

For #1 I see !, so right off the bat I'm thinking Ratio Test.

For #2, I'm not quite sure at the moment....I thought about Root Test, but not sure if that'll work...

For #3, I would try direct comparison test, comparing to 1/n^2


Try and reply back if you get stuck.

 

[skeeter]

for #2, compare to the convergent geometric series (2/e)ⁿ

 

[galactus]

2) (n>=0)∑ [(2^(n)) / (1+e^(n)]

You could use the ratio test on this one.

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

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

Using long division we get:

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

What's the limit of this?. What does it say

about convergence or divergence?.