# Homework Help: Does SUM log(1+1/n)/n converge?

1. Mar 15, 2006

Prove that $$\sum \frac{1}{n}\log(1+\frac{1}{n})$$ converges.

I tried a batch of tests, but none works. Last time Raabe's test was give to me upside down. I wonder if any other tests are give to me wrongly as well.

2. Mar 15, 2006

### durt

Have you tried comparing it to 1/n^2?

3. Mar 15, 2006

I can't get it to work out

4. Mar 15, 2006

### shmoe

Post the details of your comparison...

What bounds do you know for log(1+x)?

5. Mar 15, 2006

It's unbounded.

I figured if $$\exp(\frac{1}{n}\log(1+\frac{1}{n}))=\exp(\frac{1}{n})+\exp(\frac{1}{n})/n \rightarrow 1$$ as $$n\rightarrow\infty$$, so at least the terms go to zero, although it's insufficient.

I could use the Mercator series, but we haven't seen it yet.

Last edited: Mar 15, 2006
6. Mar 16, 2006

### HallsofIvy

The first thing that was suggested was comparison- have you tried that?

The Taylor's series for ex is 1+ x + (1/2)x2+ ... so if x is positive, ex> 1+ x and x> log(1+ x) for all x. In particular, 1/n> log(1+ 1/n) for all n.

7. Mar 16, 2006

### shmoe

It's bounded by a constant in the region you're concerned with. You are looking at log(1+x) where x=1/n, so 1<1+x<2.

A constant bound will not suffice here though, you need some indication of how fast this term is going to zero. Try the log(1+x)<x Halls mentioned, this was the one I was fishing for (one function can "bound" another).

8. Mar 16, 2006