How do you solve this limit question?

Question

Find $$\lim_{n\to \infty} n\left(\frac{1}{(n+1)^2}+\frac{1}{(n+2)^2}+\cdots+\frac{1}{(n+n)^2} \right)$$

I have tried to use definite integrals. Unfortunately, I am unable to remove the '$n$' terms from my function, due to which the function isn't being converted into $x$ terms. Any suggestions or alternatives please? (I took $\Delta x$ as $n$ and $x_i$ as in)

Oh and this is the 900,000-th question of this site. Congrats! — Gaurang Tandon, Feb 05 '18 at 14:38
Do you know how to convert a limit of sum as an integral? Can you show us what you have tried here? — Gaurang Tandon, Feb 05 '18 at 14:40
I would upvote if only the formula were properly written in MathJax. — Giuseppe Negro, Feb 05 '18 at 14:51
See https://math.stackexchange.com/questions/469885/the-limit-of-a-sum-sum-k-1n-fracnn2k2 OR https://math.stackexchange.com/questions/1851967/evaluate-lim-n-to-infty-left-frac1n-sqrtn2-1-frac2n-sq/1851968#1851968 — lab bhattacharjee, Feb 05 '18 at 15:37

score 4 · Accepted Answer · edited Feb 05 '18 at 14:50

\begin{align} \lim_{n \to \infty} n \left[ \frac{1}{(n+1)^2}+ \cdots +\frac{1}{(n+n)^2} \right] &=\lim_{n \to \infty} n \left[ \frac{1}{n^2(1+1/n)^2}+ \cdots +\frac{1}{n^2(1+n/n)^2} \right]\\&=\lim_{n \to \infty} \frac 1n\left[ \frac{1}{(1+1/n)^2}+\cdots +\frac{1}{(1+n/n)^2} \right]\\ &=\lim_{n \to \infty}\frac 1n\left[ \sum_{r=1}^n \frac{1}{(1+r/n)^2} \right]\\ &=\int_{0}^1 \frac{1}{(1+x)^2} \,{\rm d}x \end{align}

How do you solve this limit question?

1 Answers1