<s>[INST] The right way to cancel out the terms in the following telescoping series So how do I cancel and simplify the terms in the following telescopic series.
Been at it for hours, cant seem to figure it out.
$\sum\limits_{k = 1}^n \frac{1}{2(k+1)} -\frac{1}{k+2}+\frac{1}{2(k+3)} $
Any help would be deeply appreciated.
P.S: I need to show that its equal to the following,
$\frac{1}{12} - \frac{1}{2(n+2)} + \frac{1}{2(n+3)}$
But I cant seem to figure out the right cancellation method.
 [/INST] We have:
$$
a_n=\sum\limits_{k = 1}^n \left(\frac{1}{2(k+1)} -\frac{1}{k+2}+\frac{1}{2(k+3)}\right)= a_{n-1}+ \left(\frac{1}{2(n+1)} -\frac{1}{n+2}+\frac{1}{2(n+3)}\right)=a_{n-1}+\frac{1}{(n+1)(n+2)(n+3)}
$$
Does this solve your problem?
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