# Evaluate $\sum_{r=2}^{\infty} \frac{2-r}{r(r+1)(r+2)}$

Evaluate $$\sum_{r=2}^{\infty} \frac{2-r}{r(r+1)(r+2)}$$

So in a previous part of the question I calculated that $$\sum_{r=1}^{n} \frac{2-r}{r(r+1)(r+2)} = \sum_{r=1}^{n}\left( \frac{1}{r}-\frac{3}{r+1}+\frac{2}{r+2}\right)=\frac{n}{(n+1)(n+2)}$$

So my question is then why does $$\sum_{r=2}^{\infty} \frac{2-r}{r(r+1)(r+2)} = -\frac{1}{6}$$

As I though that it would be $$\frac{1}{2}$$.

Note that\begin{align}\frac{2-r}{r(r+1)(r+2)}&=\frac1r-\frac3{r+1}+\frac2{r+2}\\&=\frac1r-\frac1{r+1}-\frac2{r+1}+\frac2{r+2}.\end{align}Therefore, your series is naturally the sum of two telescoping series and its sum is$$\frac12-\frac23=-\frac16.$$

• (+1) was a minute late! – TheSimpliFire Dec 31 '18 at 15:54

Note that \begin{align}\sum_{r=2}^\infty\left( \frac{1}{r}-\frac{3}{r+1}+\frac{2}{r+2}\right)&=\sum_{r=2}^\infty\left(\frac1r-\frac1{r+1}\right)-2\sum_{r=2}^\infty\left(\frac1{r+1}-\frac1{r+2}\right)\\&=\left(\frac12-\frac13+\frac13-\frac14+\cdots\right)-2\left(\frac13+\frac14-\frac14+\frac15-\cdots\right)\\&=\frac12-\frac23=-\frac16\end{align}

• (+1) Believe me: I know what it feels to be a few seconds too late. – José Carlos Santos Dec 31 '18 at 15:55

Actually your finite sum is equal to $$\sum_{r=2}^n\frac{2-r}{r(r+1)(r+2)}=\frac{2}{n+2}-\frac{1}{n+1}-\frac{1}{6}$$

Let $$\dfrac{2-r}{r(r+1)(r+2)}=f(r)-f(r+1),$$ where$$f(m)=\dfrac{am+b}{m(m+1)}$$

$$\implies2-r=(r+2)(ar+b)-r(ar+a+b)=ar+2b$$

Set $$2b=2\iff b=1,ar=-r\iff a=-1$$

$$\implies\sum_{r=1}^n\dfrac{2-r}{r(r+1)(r+2)}=\sum_{r=1}^n\left(f(r)-f(r+1)\right)=f(1)-f(n+1)$$

Set $$n\to\infty$$ and show that $$n\to\infty f(n)=0$$

If $$H_{n}$$ is the $$n$$-th harmonic number, we have $$\sum_{r=2}^{n}\left(\frac{1}{r}-\frac{3}{r+1}+\frac{2}{r+2}\right)=\left(H_{n}-1\right)-3\left(H_{n}-1-\frac{1}{2}\right)+2\left(H_{n}-1-\frac{1}{2}-\frac{1}{3}\right)$$ $$=-1+\frac{3}{2}-\frac{2}{3}=-\frac{1}{6}.$$ Now take the limit as $$n\rightarrow+\infty$$.

By definition,

$$\sum_{r=1}^\infty a_r = \lim_{n\to\infty}\sum_{r=1}^n a_r .$$

In your case the above limit is zero. Then:

$$\sum_{r=2}^\infty a_r = \sum_{r=1}^\infty a_r - a_1 = -a_1 .$$

This reasoning is what @HenriLee was referring to in his answer.