$\displaystyle
\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!
Hi,
Multiplying $\displaystyle \frac{1}{\sqrt{k}(k+3)+k\sqrt{k+3}}$ by $\displaystyle \frac{\sqrt{k}(k+3)-k\sqrt{k+3}}{\sqrt{k}(k+3)-k\sqrt{k+3}}=1$ I get that $\displaystyle \frac{1}{\sqrt{k}(k+3)+k\sqrt{k+3}}=\frac{1}{3\sqr t{k}}-\frac{1}{3\sqrt{k+3}}$. If you sum these terms for $\displaystyle k$ such that $\displaystyle 1\leq k\leq n$ you'll see that many of them "disappear".