<s>[INST] $\sum _{n=1}^{\infty } \frac{1}{n (n+1) (n+2)}$ Understand the representation $$\sum _{n=1}^{\infty } \frac{1}{n (n+1) (n+2)}$$=$$\frac{1}{2} \left(-\frac{2}{n+1}+\frac{1}{n+2}+\frac{1}{n}\right)$$
$$s_n=\frac{1}{2} \left(\left(\frac{1}{n+2}+\frac{1}{n}-\frac{2}{n+1}\right)+\left(\frac{1}{n-1}-\frac{2}{n}+\frac{1}{n+1}\right)+\ldots +\left(1 \frac{1}{1}-\frac{2}{2}+\frac{1}{3}\right)+\left(\frac{1}{2}-\frac{2}{3}+\frac{1}{4}\right)+\left(\frac{1}{3}-\frac{1}{2}+\frac{1}{5}\right)+\left(\frac{1}{4}-\frac{2}{5}+\frac{1}{6}\right)+\left(\frac{1}{5}-\frac{1}{3}+\frac{1}{7}\right)+\left(\frac{1}{6}-\frac{2}{7}+\frac{1}{8}\right)\right)$$
Im sorry for not being able to get the term in right order, but as you can see the first two terms in the parenthesis should be the last for a accurate representation. 
$$\left(
\begin{array}{cc}
 1 & 1-1+\frac{1}{3} \\
 2 & \frac{1}{2}-\frac{2}{3}+\frac{1}{4} \\
 3 & \frac{1}{3}-\frac{1}{2}+\frac{1}{5} \\
 4 & \frac{1}{4}-\frac{2}{5}+\frac{1}{6} \\
 5 & \frac{1}{5}-\frac{1}{3}+\frac{1}{7} \\
 6 & \frac{1}{6}-\frac{2}{7}+\frac{1}{8} \\
\end{array}
\right)$$
I think that my pure understanding of the representation is what makes me confused. From the table I see that a pattern in the denominators emerge, when n=1, then (1-2+3), when n=2 then, 2-3+4. However given the part of sn where $$\frac{1}{n-1}-\frac{2}{n}+\frac{1}{n+1}$$ I fail to see how the first part of expression right above  is not undefined when n=1.
I am convinced that it is my lack of knowledge of what the representation actually means. And I am also struggling to see what cancels to give$$\frac{1}{2} \left(\frac{1}{n+2}-\frac{1}{n+1}+\frac{1}{2}\right)$$
However given that this is true I am able to understand that the answer is (1/4). But as I said it is the representation I do not get. Could someone help me as this would be very useful for my next chapter in my textbook!
 [/INST] You should write: (define $s_n$ as the sequence of partial sums of the series)
$s_n=\sum _{k=1}^{n} \frac{1}{k (k+1) (k+2)}$
$=\sum _{k=1}^{n} {\frac{1}{2} \left(-\frac{2}{k+1}+\frac{1}{k+2}+\frac{1}{k}\right)}$
$=\frac{1}{2}\left(-2\sum_{k=1}^{n}\frac{1}{k+1}+\sum_{k=1}^{n}\frac{1}{k+2}+\sum_{k=1}^{n}\frac{1}{k}\right)$ because the sums are finite. Observe then that the three sums overlap on most terms.
$=\frac{1}{2}\left(-2\sum_{k=2}^{n+1}\frac{1}{k}+\sum_{k=3}^{n+2}\frac{1}{k}+\sum_{k=1}^{n}\frac{1}{k}\right)$ by just rewriting the sum.
$=\frac{1}{2}\left(\left(-2\sum_{k=3}^{n}\frac{1}{k}-2\cdot\frac{1}{2}-2\frac{1}{n+1}\right)+\left(\sum_{k=3}^{n}\frac{1}{k}+\frac{1}{n+1}+\frac{1}{n+2}\right)+\left(\sum_{k=3}^{n}\frac{1}{k}+\frac{1}{1}+\frac{1}{2}\right)\right)$ by just rewriting the sum again.
$=\frac{1}{2}\left(-2\cdot\frac{1}{2}-2\frac{1}{n+1}+\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{1}+\frac{1}{2}\right)$ by cancelling terms.
$=\frac{1}{2}\left(-\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{2}\right)$ by cancelling terms.
Then only now, can you consider taking the limit $n\to\infty$.
Maybe you are not understanding the $\sum$ operator.
$\sum_{k=1}^{n} \frac{1}{k(k+1)(k+2)}$ is exactly $\frac{1}{\underline{1}(\underline{1}+1)(\underline{1}+2)}+\frac{1}{\underline{2}(\underline{2}+1)(\underline{2}+2)}+\dots+\frac{1}{\underline{n}(\underline{n}+1)(\underline{n}+2)}$ by definition.
And $\sum_{k=1}^{\infty} \frac{1}{k(k+1)(k+2)}$ is exactly $\lim_{n\to\infty} \sum_{k=1}^{n} \frac{1}{k(k+1)(k+2)}$ by definition.
More Examples
$$\sum_{k=1}^n a_{k+1}=a_2+a_3+\dots+a_{n+1}=\sum_{k=2}^{n+1} a_k \quad (Eq1)$$
In general, we have $\sum_{x\in S} f(x)$, which means "take all the distinct values in a finite set $S$, apply $f$ to each of them, and sum the results up". Note the results may not be distinct.
Now we can write $\sum_{k=1}^n f(k)=\sum_{k\in\mathbb{Z}\cap[1,n]} f(k)$.
If $g$ is an injective function on $S$ (distinct inputs give distinct outputs), then 
$$\sum_{x\in S} f(x)=\sum_{y\in g(S)} f(g^{-1}(y))\quad\text{, also}\quad\sum_{x\in S} f(g(x))=\sum_{y\in g(S)} f(y)$$
where $g(S)$ are all the outputs of $g$ given inputs in $S$, and $g^{-1}$ is the function which takes as input an output of $g$ and returns the input which gave that output (this is the inverse function).
So in (Eq1), we are taking $g$ to be $k\mapsto k+1$, and $S=\mathbb{Z}\cap[1,n]$. Hence $g(S)=\mathbb{Z}\cap[2,n+1]$.
Of course, you have the distributive rule: $\sum_{x\in S}af(x)=a\sum_{x\in S}f(x)$ (use induction).
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