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the sum of series :∑I/n 2 +2n fron n=1 to n=infinite is?????

the sum of series :∑I/n2+2n fron n=1 to n=infinite is?????
 

Grade:12

1 Answers

Shobhit Varshney IIT Roorkee
askIITians Faculty 33 Points
6 years ago
Dear student,

Thanks for your post.

\sum\frac{1}{n(n+2)} = \sum\frac{2}{n(n+2)} = \frac{1}{2}\sum\frac{1}{n}\frac{1}{n+2}

=\frac{1}{2} * (\frac{1}{1} - \frac{1}{3}+ \frac{1}{2}\frac{1}{4} + \frac{1}{3}\frac{1}{5}+ …........)

Now, all terms cancel out except 1 and ½, and since, n = infinity, last term tens to zero.

\Rightarrow\frac{1}{2}* (\frac{1}{1}+ \frac{1}{2})

=\frac{3}{4}

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