How I calculate the sum of the series
∞
Σ (1/n^2-4)
n=3
Note that 1/(n^2-4) = 1/4 [1/(n-2) - 1/(n+2)]
so forget about the factor of 1/4 till the end.
Now look at the series.
1/1 - 1/5 + 1/2 - 1/6 + 1/3 - 1/7 + 1/4 - 1/8 + 1/5 - 1/9 + ...
Notice that all of the fractions from 1/5 on up disappear, leaving
1/1 + 1/2 + 1/3 + 1/4 = 25/12
Finally, 1/4 * 25/12 = 25/48