Tuesday, 28 April 2015

calculus - Find limlimitsntoinftysumlimitsnk=1frac2k+1k2(k+1)2




I have to find the limit lim I tried to make it into a telescopic series but it doesn't really work out...



\lim_{n\to\infty} \sum_{k=1}^n \frac{2k+1}{k^2(k+1)^2}=\sum_{k=1}^n \left(\frac{1-k}{k^2}+\frac1{k+1}-\frac1{(k+1)^2} \right) so that is what I did using telescopic...



I said that:



\frac{2k+1}{k^2(k+1)^2}=\frac{Ak+B}{k^2}+\frac C{k+1}+\frac D{(k+1)^2} but now as I look at it.. I guess I should "build up the power" with the {k^2} too, right?


Answer



\lim_{n\rightarrow\infty}\sum^{n}_{k=1}\bigg[\frac{1}{k^2}-\frac{1}{(k+1)^2}\bigg]



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