Problem of the week #78 - September 23rd, 2013

[Jameson]

Evaluate the following sum:

\(\displaystyle \sum_{i=1}^{10}\frac{2i+1}{i^2(i+1)^2}\)
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Remember to read the POTW submission guidelines to find out how to submit your answers!

 

[Jameson]

Congratulations to the following members for their correct solutions:

1) hxthanh
2) Chris L T521
3) MarkFL
4) eddybob123
5) johng
6) anemone

Solution (from hxthanh):

Spoiler

\begin{align*}S_n=\sum_{i=1}^n \dfrac{2i+1}{i^2(i+1)^2}&=\sum_{i=1}^n \dfrac{(i+1)^2-i^2}{i^2(i+1)^2}\\&=\sum_{i=1}^n \left(\dfrac{1}{i^2}-\dfrac{1}{(i+1)^2}\right)\\&=\dfrac{1}{1^2}-\dfrac{1}{(n+1)^2}\end{align*}

With $n=10$, we get: $S_{10}=1-\dfrac{1}{121}=\boxed{\dfrac{120}{121}}$