I was asked about convergence of summation of $\log(n)/n^2$ and $\log(1+(1/n))/n^2$. I want to use only p series test. Is there any approximation for $\log(x)$?, Like sterling's approximation for factorials. Or is there any other method?
[Math] convergence of the series involving log function
logarithmssequences-and-series
Best Answer
By p-series test,
$$\sum\frac{1}{n^{1.5}} \quad \text{and} \quad \sum\frac{1}{n^3} \quad \text{converge}$$
By the L'Hospital's rule, we have also have the following limits
$$\lim_{n\rightarrow \infty} \frac{\frac{\ln(n)}{n^{2}}}{\frac{1}{n^{1.5}}}=\lim_{n\rightarrow \infty} \frac{\ln(n)}{n^{0.5}} = 0$$ and $$\lim_{n \rightarrow \infty} \frac{\frac{\ln(1+\frac{1}{n})}{n^2}}{\frac{1}{n^3}}= \lim_{n \rightarrow \infty} \frac{\ln(1+\frac{1}{n})}{\frac{1}{n}}=1$$
By limit comparison test,
$$ \sum \frac{\ln(n)}{n^2} \quad \text{and} \quad \sum\ \frac{\ln(1+\frac{1}{n})}{n^2} \quad \text{converge}$$