Find the summation of the series: $\sum_{n=1}^{5}\frac{1}{(2n-1)(2n+1)(2n+3)}$

Question

Find the summation of the series $$\sum_{n=1}^{5}\frac{1}{(2n-1)(2n+1)(2n+3)}$$

I want hints to solve this problem.

It is a finite sum, but it is an entrance based question and hardly they allow calculators.

Thanks in advance!

@riicheyv It's a finite series, there are only 5 terms, so you can put it in the calculator? It's not very elegant, I admit, otherwise Claude's hint would be a suggestion. — imranfat, May 01 '15 at 15:34
It's an entrance based question and hardly they allow calculators. @imranfat. — gaufler, May 01 '15 at 15:41
Somewhat similar question: Find the sum of the series $\sum \frac{1}{n(n+1)(n+2)} — Martin Sleziak, May 01 '15 at 15:41
I am voting to leave open since there is an explicit request for hints. — N. F. Taussig, May 01 '15 at 16:26

score 5 · Accepted Answer · answered May 01 '15 at 15:21

5

Hint: Compute $$ \frac1{(2n-1)(2n+1)}-\frac1{(2n+1)(2n+3)} $$ and think Telescoping Series.

answered May 01 '15 at 15:21

robjohn

1

Oh... I didn't see it was a finite series. I will leave this answer in case someone is interested in the infinite series, or a sum to a variable upper limit. – robjohn May 01 '15 at 15:23

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