Struggling with Series Convergence Tests?

[G01]

Hey everyone, I have some problems here involving the comparision test and Alternating series tests for series. I've solved most but there are about five I'm lost on. All I'm asking for is a couple hints. Thanks there's only one alternating series question so I'll put that first.

AST:
$$\sum^{\infty}_{n=1} (-1)^nsin(\pi/n)$$

Comparison or Limit Comparison:

$$\sum^{\infty}_{n=2} \frac{n^2+1}{n^3-1}$$

$$\sum^{\infty}_{n=1} \frac{1}{n!}$$

$$\sum^{infty}_{n=1} \sin(\frac{1}{n})$$

OK and this last one :

$$\sum^{\infty}_{n=1} \frac{5+2n}{(1+n^2)^2}$$
I know I don't have any work but I'm stuck on these five. All I'm asking for is a couple hints thanks in advance.

 

[StatusX]

You can do a little work. Do you understand these methods? Where exactly are you getting stuck?

 

[G01]

Alright for the comparison or limit test problems I get stuck trying to find the series to compare it too. Is there any advice that makes picking a comarison series easier? I'l work on these a little more and post what I get.

 

[G01]

Yeah I got the answer for the first comparison test question, that is divergent I ended up comparing it to N^2/N^3.

 

[StatusX]

OK, that works. The same idea for works for the last one. For the sin(1/n) one, consider what sin(x) looks like as x->0. For the n! one, think geometric series.

 

[G01]

OK, got the last one. I multiplied out the denominator and then found a comparison. Still stuck on the sin and factorial ones tough.

 

[StatusX]

So... do you know what sin(x) looks like as x->0 or what a geometric series is?