Does SUM log(1+1/n)/n converge?

[Treadstone 71]

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.

 

[durt]

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

 

[Treadstone 71]

I can't get it to work out

 

[shmoe]

Post the details of your comparison...

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

 

[Treadstone 71]

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.

 

[HallsofIvy]

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

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

 

[shmoe]

It's unbounded.

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).

 

[Treadstone 71]

I got it. Thanks for the help.