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How to find the sum of the following infinite series?

$$\frac1{2 \cdot 3 \cdot 4}+\frac1{4 \cdot 5 \cdot 6}+\frac1{6 \cdot 7 \cdot 8}+\cdots$$

I can write its partial sum but am unable to proceed.

jvdhooft
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laxman kumar
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  • Multiplication is something different from the end of sentence, and so it has its own symbol, different from a full stop aka period. You can obtain it with the LaTeX command \cdot, like in 2 \cdot 3 which is rendered as $2 \cdot 3$. – CiaPan Jun 05 '18 at 06:37

1 Answers1

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$$\frac1{(n-1)n(n+1)}=\frac12\left(\frac1{n-1}-\frac2n+\frac1{n+1}\right)$$ so your sum is $$\frac12\left(\frac12-\frac23+\frac24-\frac25+\frac26-\frac27+\cdots\right).$$ This is very similar to $$\frac11-\frac12+\frac13-\frac14+\cdots.$$

Angina Seng
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