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We have the infinite series:$$\frac{1}{2\cdot 3\cdot 4}+\frac{1}{4\cdot 5\cdot 6}+\frac{1}{6\cdot 7\cdot 8}+\cdots$$

This is not my series: $\frac{1}{1\cdot 2\cdot 3}+\frac{1}{2\cdot 3\cdot 4}+\frac{1}{3\cdot 4\cdot 5}+\frac{1}{4\cdot 5\cdot 6}+\cdots$ so I cannot use $\sum_{k=1}^n \frac{1}{k(k+1)(k+2)}$

My attempt:

I know that this type of series solved by making it telescoping series but here, I am unable find general term of the series. Thank you.

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    http://math.stackexchange.com/questions/2108596/formula-for-sum-k-1n-frac1kk1k2 – 2017-01-23
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    My series is not like this:$\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+\frac{1}{4.5.6}+\cdots$. – 2017-01-23
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    I think to find $\frac{1}{2\cdot3\cdot4}+\frac{1}{5\cdot6\cdot7}+\frac{1}{8\cdot9\cdot10}+...$ is much interesting. – 2017-01-23

3 Answers 3

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$$\dfrac2{2n(2n+1)(2n+2)}=\dfrac{2n+2-2n}{2n(2n+1)(2n+2)}=\dfrac1{2n(2n+1)}-\dfrac1{(2n+1)(2n+2)}$$

$$=\dfrac1{2n}-\dfrac1{2n+1}-\left(\dfrac1{2n+1}-\dfrac1{2n+2}\right)$$

$$\sum_{n=2}^\infty\left(\dfrac1{2n}-\dfrac1{2n+1}-\left(\dfrac1{2n+1}-\dfrac1{2n+2}\right)\right)$$

$$=\left(\dfrac12-\dfrac13+\dfrac14-\dfrac15+\cdots\right)-\left(\dfrac13-\dfrac14+\dfrac15+\cdots\right)$$

Now $\ln2=1-\dfrac12+\dfrac13-\dfrac14+\cdots$

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    It's $3/2-\log4$right? – 2017-01-23
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    @lab bhattacharjee: I think that from line 3 to 4, you changed the order of summation on an infinite number of terms: you sum $\frac 1{2n} - \frac 1 {2n+1}$ on one side, and $\frac{1}{2n+1} - \frac 1 {2n+2}$ on the other. However, this is not legit in the general case (see [Riemann series theorem](https://en.wikipedia.org/wiki/Riemann_series_theorem)) – 2017-01-23
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    @Mariuslp:then according to you what will be the answer$?$ – 2017-01-23
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    @MatheMagic: My bad, I answered too quickly. This answer is indeed correct. – 2017-01-24
2

Let $$ f(x)=\sum_{n=1}^\infty\frac1{n(n+1)(n+2)}x^{n+2}. $$ Then $$ f'(x)=\sum_{n=1}^\infty\frac1{n(n+1)}x^{n+1},f''(x)=\sum_{n=1}^\infty\frac1{n}x^{n},f'''(x)=\sum_{n=1}^\infty x^{n-1}=\frac1{1-x}. $$ So $$ f''(x)=-\ln(1-x)$$ and $$ f(1)=-\int_0^1\int_0^x\ln(1-t)dtdx=\cdots$$

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We have,

$$\sum_{k=1}^{n} \frac{1}{2k(2k+1)(2k+2)}$$

Hope you can solve it further.