Hi, I'm not able to check if the following serie converges or not: $\displaystyle \sum_{n} \frac{1}{\sqrt{n(1+n^2)}}$ The ratio test didn't work, and I think I need a majorant or a minorant but I can't find one.
Follow Math Help Forum on Facebook and Google+
Originally Posted by Siron Hi, I'm not able to check if the following serie converges or not: $\displaystyle \sum_{n} \frac{1}{\sqrt{n(1+n^2)}}$ Use limit comparison with the convergent p-series $\displaystyle \sum_n\frac1{n^{3/2}}$. Spoiler: $\displaystyle \lim_{n\to\infty}\frac{1/\sqrt{n+n^3}}{1/n^{3/2}}$ $\displaystyle =\lim_{n\to\infty}\frac{n^{3/2}}{\sqrt{n+n^3}}$ $\displaystyle =\lim_{n\to\infty}\sqrt{\frac{n^3}{n+n^3}}$ $\displaystyle =\lim_{n\to\infty}\sqrt{\frac1{\frac1{n^2}+1}}$
Thanks!
View Tag Cloud