A limit of a sequence when squeeze doesn't work

Question

$$\lim_{n\to\infty}\sum_{k=1}^{n}{{1}\over{\sqrt{n^2+2kn}}}$$

How should one approach such a question? A hint would be helpful

Check the question , It is very similiar http://math.stackexchange.com/questions/266322/how-to-find-limit-of-the-sequence-sum-limits-k-1n-frac1-sqrt-n2-kn/266325#comment581562_266325 — Mathlover, Jan 28 '13 at 09:00

score 5 · Accepted Answer · edited Jan 28 '13 at 08:56

5

Hint: rewrite this as a Riemann sum, so that the limit is an integral.

edited Jan 28 '13 at 08:56

Jonas Meyer

53,602

answered Jan 28 '13 at 08:55

Thomas Andrews

177,126

score 2 · Answer 2 · answered Feb 12 '13 at 09:31

2

I try
\begin{align}\lim_{n\to\infty}\sum_{k=1}^{n}{{1}\over{\sqrt{n^2+2kn}}}&=\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}{{1}\over{\sqrt{1+\frac{2k}{n}}}}\\ &=\int_{0}^1\frac{1}{\sqrt{1+2x}}\,\,dx\end{align}

answered Feb 12 '13 at 09:31

DeeJay

211

A limit of a sequence when squeeze doesn't work

2 Answers2