# Thread: Series Help(Limit And Submation)

1. ## Series Help(Limit And Submation)

$
\lim_{n\to\infty} \sum^{n}_{k=1} \frac{1}{\sqrt{k}(k+3)+k\sqrt{k+3}}
$

please tell me the calculating and Thankyou!

2. Hi,
Originally Posted by frercss
$
\lim_{n\to\infty} \sum^{n}_{k=1} \frac{1}{\sqrt{k}(k+3)+k\sqrt{k+3}}
$
Multiplying $\frac{1}{\sqrt{k}(k+3)+k\sqrt{k+3}}$ by $\frac{\sqrt{k}(k+3)-k\sqrt{k+3}}{\sqrt{k}(k+3)-k\sqrt{k+3}}=1$ I get that $\frac{1}{\sqrt{k}(k+3)+k\sqrt{k+3}}=\frac{1}{3\sqr t{k}}-\frac{1}{3\sqrt{k+3}}$. If you sum these terms for $k$ such that $1\leq k\leq n$ you'll see that many of them "disappear".