strategies to find explicit formulae for series I have been manipulating a certain series for several hours without finding any pattern. Hence I am wondering what some of the better strategies are to find patterns and thus an explicit formula for a series. Among the things I have tried so far are:


*

*looking for a common difference between terms

*looking for a common ratio between terms

*reversing the order of the terms and summing them up, to check whether the result will be the same for all terms

*bringing the terms to a common denominator and looking for a obvious pattern in the numerator
I had no luck with any of these and others. The series btw. is $\sum_{k = 1}^n\frac{k - 1}{k(k + 1)(k + 2)}$
This is oen of the things I tried:
$\begin{align*}
                    S_n & = \frac{0}{6} + \frac{1}{24} + \frac{2}{60} + \frac{3}{120} + \frac{4}{210} + \frac{6}{336}\\
                        & = \frac{0}{6} + \frac{1}{24} + \frac{1}{30} + \frac{1}{40} + \frac{1}{52,5} + \frac{1}{56}\\
                        & = \frac{0}{1680} + \frac{70}{1680} + \frac{56}{1680} + \frac{42}{1680} + \frac{32}{1680} + \frac{30}{1680}\\
        a_n - a_{n+1} : &  -\frac{70}{1680}; \frac{14}{1680}; \frac{14}{1680}; \frac{10}{1680}; \frac{8}{1680}; 
\end{align*}$
The differences between the terms get ever smaller and the sum approaches $.25$, but any internal pattern remains hidden after the things I tried. So, are there a number of useful methods to uncover patterns in series?

Hints:


*

*$\require{cancel}\;\sum_{k = 1}^n\frac{k - 1}{k(k + 1)(k + 2)} = \sum_{k = 1}^n\frac{\cancel{k}}{\cancel{k}(k + 1)(k + 2)} - \sum_{k = 1}^n\frac{1}{k(k + 1)(k + 2)}$

*$\;\sum_{k = 1}^n\frac{1}{(k + 1)(k + 2)} = \sum_{k = 1}^n\left(\frac{1}{k + 1} - \frac{1}{k + 2}\right)$

*$\;\sum_{k = 1}^n\frac{1}{k(k + 1)(k + 2)} = \frac{1}{2}\sum_{k = 1}^n\left(\frac{1}{k(k+1)} - \frac{1}{(k+1)(k+2)}\right)$
