2006 SMT/Team Problems/Problem 8: Difference between revisions
Created page with "==Solution== To begin, let's rewrite the sum as <cmath>\lim_{n\to\infty} \sum_{k=n^2}^{(n+1)^2} \frac{1}{\sqrt{k}} = \lim_{n\to\infty} \sum_{k=0}^{2n+1} \frac{1}{\sqrt{n^2+..." |
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==Problem== | |||
Evaluate: <math> \lim_{n\to\infty}\sum_{k=n^2}^{(n+1)^2}\frac{1}{\sqrt{k}} </math> | |||
==Solution== | ==Solution== | ||
![\[\lim_{n\to\infty} \sum_{k=n^2}^{(n+1)^2} \frac{1}{\sqrt{k}} = \lim_{n\to\infty} \sum_{k=0}^{2n+1} \frac{1}{\sqrt{n^2+k}}\]](http://latex.artofproblemsolving.com/e/a/d/ead93d06f86f666c47c74cb7799102083eb06ede.png)
![\[\lim_{n\to\infty} \sum_{k=0}^{2n+1}\frac{1}{\sqrt{n^2+k}}=\lim_{n\to\infty} \sum_{k=n^2}^{(n+1)^2} \frac{1}{\sqrt{k}} = \boxed{2}\]](http://latex.artofproblemsolving.com/c/0/d/c0da643ee374f6e529942896c4592181dd560785.png)
