given the following sum : $\sum_{n=1}^{\infty}\frac{1}{(n+1)(n+2)}$ determine if convergent or divergent.
we already solved it in class, however after reviewing it later, I stumbled upon this particular nuance: we decomposed it into partial fractions as follows: $\frac{1}{(n+1)(n+2)}=\frac{A}{n+1}+\frac{B}{n+2}$
$1=A(n+2)+B(n+1)$ then we just substituted n=-1, n=-2(values that cancel 'A' or 'B' accordingly) ,found A & B(A=1, B=-1) and then rewrite the new therm, and solved it(Answer: the series is convergent and the sum is 0.5).
My point is: I'm not looking for a way to solve it, just to clarify why is it "legal" to susbstitute n=-1 and n=-2 in the partial decomposition section if we are starting to sum the series from index(in this case letter 'n') from n=1?