Convergence of sum of (3n²+5n)/(2ⁿ(n²+1))

[Telemachus]

Homework Statement

Well, hi there. I have to study the convergence of the next series using the comparison criteria, or the comparison criteria through the limit of the quotient.

$$\displaystyle\sum_{n=1}^\infty{\displaystyle\frac{3n^2+5n}{2^n(n^2+1)}}$$

I think that I should use the comparison criteria through the limit of the quotient, so I think of using for this this other series:
$$\displaystyle\sum_{n=1}^\infty{\displaystyle\frac{3n^2}{2^n}}$$

But I don't know how to determine the convergence/divergence of this series neither. I need some help.

Thanks!

 

[icystrike]

Tried using ratio test? :)

 

[Dick]

Homework Statement

Well, hi there. I have to study the convergence of the next series using the comparison criteria, or the comparison criteria through the limit of the quotient.

$$\displaystyle\sum_{n=1}^\infty{\displaystyle\frac{3n^2+5n}{2^n(n^2+1)}}$$

I think that I should use the comparison criteria through the limit of the quotient, so I think of using for this this other series:
$$\displaystyle\sum_{n=1}^\infty{\displaystyle\frac{3n^2}{2^n}}$$

But I don't know how to determine the convergence/divergence of this series neither. I need some help.

Thanks!

Two words. Ratio test.

 

[Telemachus]

Thanks. But the thing is that it asks me to do it using a comparison criteria :P

 

[jgens]

$$\frac{3n^2+5n}{2^n(n^2+1)} \leq \frac{3n^2+5n}{2^nn^2} = \frac{3n^2}{2^nn^2} + \frac{5n}{2^nn^2}$$

Now, what can you say about the convergence of the two terms on the right?

 

[Telemachus]

Thanks. They converge :D