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 !
Well if you do the limit comparison you will need to calculate the limit
$\displaystyle \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.
Here is a link to the theorem:
L'Hôpital's rule - Wikipedia, the free encyclopedia
$\displaystyle \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
$\displaystyle \lim_{n \to \infty}\frac{\frac{1}{ 1+\frac{1}{n}}\cdot \frac{-1}{n^2}}{\frac{-1}{n^2}}=1$