sequences and series of complex numbers

A sequence of complex numbers is just an ordered list z_1, z_2, z_3, ... where each z_n lives in the plane rather than on the line. We ask the same question we asked in real calculus: as n grows, do these dots settle down toward some single point z, or do they wander forever? A series is what you get when you decide to add the dots up: the infinite sum z_1 + z_2 + z_3 + ... means the running totals (the partial sums s_N = z_1 + ... + z_N) themselves form a sequence, and the series 'equals' the limit of those running totals if that limit exists.

Convergence in the plane is wonderfully simple because it splits into two ordinary real problems. Write z_n = x_n + i y_n. Then z_n converges to z = x + i y exactly when x_n converges to x AND y_n converges to y — the real and imaginary parts each converge on the number line. Equivalently, z_n converges to z when the distance |z_n - z| shrinks to 0. So everything you know about real limits transfers; the only new flavour is that 'close' now means close as points in the plane, measured by the modulus. A useful sanity check: if the series sum z_n converges, then its terms z_n must shrink to 0 (the n-th term test). That is necessary but, as on the real line, not sufficient.

These ideas are the foundation for everything that follows in this part of the subject. A power series — the local DNA of a holomorphic function — is exactly an infinite series whose terms are complex numbers a_n (z - z_0)^n, one for each fixed z. Asking where such a series converges is asking, for which z does this complex series settle down? So sequences and series are not a dry preliminary; they are the language in which Taylor's theorem and analyticity will be spoken.