summation from n=1 to infinity ((n+1)^{0.5}-(n)^{0.5})/(n^{2}+n)^{0.5}.
I've tried using limit comparison test but the ratio i got approaches 0 and thus no conclusion can be drawn.
summation from n=1 to infinity ((n+1)^{0.5}-(n)^{0.5})/(n^{2}+n)^{0.5}.
I've tried using limit comparison test but the ratio i got approaches 0 and thus no conclusion can be drawn.
note ...
$\dfrac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n^2+n}} \cdot \dfrac{\sqrt{n+1} + \sqrt{n}}{\sqrt{n+1} + \sqrt{n}} = \dfrac{1}{\sqrt{n^2+n} (\sqrt{n+1} + \sqrt{n})} < \dfrac{1}{\sqrt{n^3}}$
what can you say about the series, $\displaystyle \sum \dfrac{1}{n^{3/2}}$ ... ?