Question concerning Infinite Series

Problem : The collection $\sum_{n=1}^{\infty} a_n$ deviate and also favorable. What can be claimed on the collection $\sum_{n=1}^{\infty} \frac{a_n}{1+n^{2}a_n}$?

Strategy :

  1. First, I attempted dividing it right into 2 instances : $a_n \to \infty$, $a_n \to A$ (where $A$ is some const value).

  2. Attempted the proportion examination and also obtained no place with that said.

I assume that the appropriate strategy is to examine each instance independently, yet my suspicion informs me that there is a workaround.

2019-12-02 02:54:35
Source Share
Answers: 2

Hint : Rewrite as $$\sum_{n=1}^{\infty} \frac{1}{\frac{1}{a_n} + n^2}.$$

2019-12-03 04:26:47

Divide the numerator and also by $a_n$, and also contrast what you get with $1/n^2$.

2019-12-03 04:26:42