Does this series converge or diverge?

[ghost34]

Homework Statement

does the following series converge or diverge? how does one determine whether it diverges or converges?

Homework Equations

Ʃ (n+1)/((n^3)+3n^2+5)^1/2

n=1, to infinity

The Attempt at a Solution

I attempted to compare it with n/(n)^3/2, this series diverges and is greater than the original series, so the comparison test didn't work, at least in this case. the ratio test was inconclusive as well. any help would be appreciated, thanks

 

[Dansuer]

Try using the asymptotic comparison test. I'll write it down in case you don't know it.


If |a_{n}| ~ |b_{n}|

then

\sum |a_{n}| converges \Leftrightarrow \sum |b_{n}| converges


~ means "is asymptotic to".

a_{n} ~ b_{n} \Leftrightarrow limit {a_{n}}/{b_{n}}\rightarrow 1

It's a very very useful test.

 

[vela]

Dansuer's suggestion is good, but you probably know it as the limit comparison test. The limit has to be non-zero and finite, not necessarily equal to 1.

 

[HallsofIvy]

The crucial point is that the denominator has "leading power" 3/2 while the numerator has power 1. The entire fraction has "power" -1/2. You should be able to compare it with 1/n^{1/2}.