a partial-fraction expansion
From algebra you know that a rational function can be broken into simple pieces — for example 1/(z^2 - 1) splits as (1/2)/(z - 1) - (1/2)/(z + 1), one fraction per pole. A partial-fraction expansion is this same idea pushed to a meromorphic function with infinitely many poles: you write it as a sum of one simple term per pole (its principal part there), gathered into a convergent series. Where a power series displays a function near a single point, a partial-fraction expansion displays it across all its poles at once.
Concretely, if f is meromorphic with simple poles at b_1, b_2, ... and residue r_n at b_n, the heart of the expansion is the sum over n of r_n / (z - b_n) — each term is exactly the principal part at one pole. This raw sum often diverges, so (following Mittag-Leffler) you subtract convergence-correcting constants and possibly add an entire function; the polished result is a series that converges to f on compacta away from the poles. To read off the terms in practice: the residue r_n is found exactly as in residue calculus, e.g. for a simple pole r_n = lim as z -> b_n of (z - b_n) f(z).
These expansions are powerful in two directions. Read left to right, they let you understand a complicated meromorphic function as a transparent sum of poles. Read right to left, they are a machine for summing series: setting z to a convenient value in an identity like the cotangent expansion collapses an infinite sum of simple fractions into a closed form, which is how Euler-style evaluations such as sum 1/n^2 = pi^2/6 fall out. So partial fractions are simultaneously a structural decomposition and a calculational tool.
The function pi^2 / sin^2(pi z) has a double pole at every integer with principal part 1/(z - n)^2. Its partial-fraction expansion is pi^2 / sin^2(pi z) = sum over all integers n of 1/(z - n)^2. This sum converges with no corrections needed (the terms decay like 1/n^2), and it lays the whole function out as a clean sum over its double poles.
pi^2 / sin^2(pi z) as a sum of 1/(z - n)^2 over the integers — a partial-fraction expansion that converges on its own.
Whether you need the correcting subtractions depends on how fast the principal parts decay: for double poles with terms like 1/(z-n)^2 the bare sum already converges, but for simple poles 1/(z-n) it does not, and corrections are mandatory. Skipping them gives a divergent, meaningless 'expansion'.