Thread: Convergence/divergence of the series (a_i / (1+a_i))

1. Convergence/divergence of the series (a_i / (1+ia_i))

I need help trying to determine the convergence/divergence of $\sum_{i=1}^{\infty} \frac{a_i}{1 + ia_i}, a_i>0$. I thought about comparison test, but I can't find anything to compare this series to.

EDIT: Might integrals help?

2. Re: Convergence/divergence of the series (a_i / (1+a_i))

Originally Posted by phys251
I need help trying to determine the convergence/divergence of $\sum_{i=1}^{\infty} \frac{a_i}{1 + ia_i}, a_i>0$. I thought about comparison test, but I can't find anything to compare this series to.
The series in the title is different from from the one in the post.
Which is correct?

3. Re: Convergence/divergence of the series (a_i / (1+ia_i))

Originally Posted by Plato
The series in the title is different from from the one in the post.
Which is correct?
The post, not the title. a_i / (1 + a_i) is pretty easy to deal with. The other form, a little tougher.

4. Re: Convergence/divergence of the series (a_i / (1+ia_i))

Originally Posted by phys251
The post, not the title. a_i / (1 + a_i) is pretty easy to deal with. The other form, a little tougher.

There is no clear answer, $\sum_{i=1}^{\infty} \frac{a_i}{1 + ia_i}, a_i>0$

If $a_i=\frac{1}{i}$ it clearly diverges.

If $a_i=\frac{1}{2^i}$ it clearly converges.

Have you given the entire question?

5. Re: Convergence/divergence of the series (a_i / (1+a_i))

I am also given that $\sum_{i=1}^{\infty} a_i$ diverges. No information on $a_i$ except that $a_i$ is always positive.