Sequences & Series

When asked to sum a finite or infinite series whose term has no obvious closed form, try to write that term as $g(r)-g(r+1)$ (or $g(r)-g(r+k)$): the sum then collapses to the surviving endpoints $g(1)-g(N+1)$, and the infinite sum is $g(1)-\lim g(N+1)$.

Trigger: a sum to evaluate exactly, especially a product in the denominator, a $\tan /\arctan$ term, or a difference you are handed in a 'by considering ...' hint.

Instances:
(i) partial fractions on a rational term, e.g. $\frac{1}{r(r+1)(r+2)}$ or $\frac{2}{r(r^2-1)}$, split so consecutive pieces cancel;
(ii) a trigonometric term turned into a difference by a product-to-sum identity, e.g. $2\sin \frac{1}{2}x\cos rx=\sin (r+\frac{1}{2})x-\sin (r-\frac{1}{2})x$, so $\sum \cos rx$ telescopes;
(iii) an $\arctan$ term written as $\arctan u_r-\arctan u_{r+1}$ using $\arctan a-\arctan b=\arctan \frac{a-b}{1+ab}$;
(iv) a factored difference handed to you, $\frac{1}{1+x^r}-\frac{1}{1+x^{r+1}}$, whose terms you recognise as the summand.