# Series Convergence

1. Oct 17, 2005

Given $$a_{n} > 0$$ and $$\sum a_{n}$$ diverges, show that $$\sum \frac{a_{n}}{1+a_{n}}$$ diverges.
Since I don't have an explicit form for the series, I can't apply any of the standard tests. I'm not sure where to start on this problem. I know the criteria for convergence/divergence, namely tail end of series has to converge or cauchy criterion condition. But I don't see how that helps without knowing what series looks like. Please steer me in the right direction.

2. Oct 18, 2005

### Fermat

If a series diverges, what happens to to its reciprocal ?

3. Oct 18, 2005

I would say reciprocal converges, but apparently it's not enough to bring original series to convergence... I thought about this a little more and I think I'll analyze it based on how $$a_{n}$$ diverges. that is, does it go to zero, constant, or infinity as n goes to infinity and try to bound the reciprocal from below to show series diverges.

4. Oct 18, 2005

### amcavoy

For $a_n>1$:

$$\frac{a_n}{1+a_n}>\frac{1}{2}$$

For $a_n\leq 1$:

$$\frac{a_n}{1+a_n}\geq\frac{a_n}{2}$$