1. ## Converge or Diverge

Here is a nasty series:

Pn = (n+1)2^n/(n!)^2

Using the ratio test I get:

Pn+1/Pn = 2/(n+1)(n!)

However I dont think this is correct, plus I dont even Know if this is the best way to test if the above nasty is convergent

and Im going to cry because I'm stuck

2. Originally Posted by partyshoes
Here is a nasty series:

Pn = (n+1)2^n/(n!)^2

Using the ratio test I get:

Pn+1/Pn = 2/(n+1)(n!)

However I dont think this is correct, plus I dont even Know if this is the best way to test if the above nasty is convergent

and Im going to cry because I'm stuck

using the ratio test is fine. but that's wrong.

By the ratio test: $\lim_{n \to \infty} \left| \frac {P_{n + 1}}{P_n} \right| =$ $\lim_{n \to \infty} \left| \frac {\frac {(n + 2)2^{n + 1}}{[(n + 1)!]^2}}{\frac {(n + 1)2^n}{(n!)^2}} \right|$ $= \lim_{n \to \infty} \left| \frac {(n + 2)2^{n + 1}}{[(n + 1)!]^2} \cdot \frac {(n!)^2}{(n + 1)2^n} \right|$

now continue

3. Cancelled out the 2^n but still left with;

2(n+2)n!^2/(n+1)!^2(n+1)

What can I do to simplify this?

4. Originally Posted by partyshoes
Cancelled out the 2^n but still left with;

2(n+2)n!^2/(n+1)!^2(n+1)

What can I do to simplify this?
what you wrote is confusing me. your syntax is off.

start by grouping the common things. that is, put the factorial over the factorial, the 2^(whatever) over the 2^(whatever) and so on. you end up with

$\lim_{n \to \infty} \left| \frac {n + 2}{n + 1} \cdot \frac {2^{n + 1}}{2^n} \cdot \left( \frac {n!}{(n + 1)!} \right)^2 \right|$

now can you continue?

5. Does this simplify to:

2(n+2)/(n+1)^3

6. Originally Posted by partyshoes

Does this simplify to:

2(n+2)/(n+1)^3
yes. now what does the limit of that function go to? does it converge?