Math Help - convergence test

1. convergence test

I have question which to show that sum(1/n)log[1+(1/n)] is convergent. I don't think i can use integral or ratio or raabe's test. I am guessing I have to use limit comparison test ??? Help me !

2. Originally Posted by mathsohard
I have question which to show that sum(1/n)log[1+(1/n)] is convergent. I don't think i can use integral or ratio or raabe's test. I am guessing I have to use limit comparison test ??? Help me !
Hint: compare to

$\frac{1}{n^2}$

and use l'Hospitals rule

3. I haven't learned l'Hospitals rule yet lol is that the only way ???

4. Originally Posted by mathsohard
I haven't learned l'Hospitals rule yet lol is that the only way ???
Well if you do the limit comparison you will need to calculate the limit

$\frac{\frac{\log\left( 1+\frac{1}{n}\right)}{n}}{\frac{1}{n^2}}=n \log \left( 1+\frac{1}{n} \right)$

This is an indeterminate form of infinity times zero. What methods have you learned to resolve these types of limits? The only other method I can think of is to use a Taylor series expansion, but that is usually covered after the topics you are learning right now.

5. Well can you show me how to do with l'Hospitals rule then???

6. Originally Posted by mathsohard
Well can you show me how to do with l'Hospitals rule then???
Here is a link to the theorem:

L'Hôpital's rule - Wikipedia, the free encyclopedia

$\frac{\frac{\log\left( 1+\frac{1}{n}\right)}{n}}{\frac{1}{n^2}}=\frac{ \log \left( 1+\frac{1}{n} \right)}{\frac{1}{n}}$

This is of the form zero over zero so we take the derivative of the numerator and denominator to get

$\lim_{n \to \infty}\frac{\frac{1}{ 1+\frac{1}{n}}\cdot \frac{-1}{n^2}}{\frac{-1}{n^2}}=1$