Convergence/Divergence?

Question

$$\sum\ n{((2/5)^n + n^2)}.$$

I was told this was possible with the limit comparison test, but most examples using LCT are with fractions. Is it still possible to use it and how?

or what is a better way to solve this? I would prefer if it wasnt with Divergence Test

Answer 1

The general term of the series given by $$u_n=n ((\frac {2}{5})^n+n^2)$$ is equivalent to $n^3$ thus it is divergent.

Answer 2

Whether multiplication or division, the sum diverges.

If multiplication, obviously, since individual terms go to infinity.

If division, with the terms being $\dfrac{n}{((2/5)^2 + n^2)} $, they are like $\dfrac1{n}$, and the sum of this diverges.

Answer 3

Also, you can notice that if we define

$$a_n = n \left( \left(\frac{2}{5} \right)^n+n^2 \right)$$ then the limit of $a_n$ as $n \rightarrow \infty$ is $\infty \neq 0$, so the series has to diverge, since the terms of the series are getting aribtarily large.