Converging Series Homework: Can't Prove It!

[Mattofix]

Homework Statement

I have done really well previous to these questions but i don't have a clue where to start any of them.

Do these converge?

i) Sum from n=1 to infinty of sin(2^n)/2^n

ii) Sum from n=1 to infinty of ncosn/(n^3 + logn)

iii) Sum from n=1 to infinty of (logn)^4/n^2

Homework Equations

Use comparison test but i can't get it them in a simple form or definitely prove it.

The Attempt at a Solution

i) yes

ii)yes

iii)unsure

No proof though...hmmmm...

 

[Dick]

You must have some reason for the yes answer on i) and ii). What is it? Remember in the comparison test you can do anything to the numerator that makes it larger and anything to the denominator that makes it smaller (in absolute value). If the resulting series converges, so does the original series. Now what's a nice round number that is greater than sin(2^n)?

 

[Mattofix]

beutiful man - got that one - how about the other 2?

 

[Dick]

Same trick n/n^3>cos(n)*n/(n^3+log(n)). The third is a little trickier. Do you know that log(n)/n^p goes to zero as n->infinity for all p>0?