sequences and convergence in the plane
A sequence of complex numbers is just an endless list z_1, z_2, z_3, ... of points in the plane. It converges to a limit L if the points eventually huddle arbitrarily close to L — precisely, if for every epsilon > 0 there is an index N beyond which |z_n - L| < epsilon. Geometrically, every disk around L, however small, contains all the terms of the sequence from some point on. If no such L exists, the sequence diverges.
Convergence in the plane is, comfortingly, just two real convergences at once. Write z_n = x_n + i y_n. Then z_n converges to L = a + i b if and only if x_n converges to a and y_n converges to b as real sequences. This follows from the same modulus inequalities as for continuity: the distance |z_n - L| is small exactly when both coordinate-distances are small. So you never need a new theory of convergence — you reuse real sequences componentwise.
Sequences quietly underpin a great deal. The whole apparatus of series (the partial sums form a sequence), of power series, of Taylor and Laurent expansions, and of limits of functions, rests on convergence in the plane. A key supporting fact is completeness: the plane has no missing points, so a Cauchy sequence (whose terms eventually get close to each other) always converges to an actual point. Combined with boundedness, this gives the Bolzano-Weierstrass property — every bounded sequence has a convergent subsequence — the seed of countless compactness and normal-family arguments.
The sequence z_n = (1 + i)/n converges to 0: its real part 1/n and imaginary part 1/n both go to 0. The sequence z_n = i^n = i, -1, -i, 1, i, ... does not converge — it cycles through four points forever.
z_n converges if and only if its real and imaginary parts each converge as real sequences.
Convergence here is convergence in modulus (distance), the only natural notion in the plane. There is no 'order' on the complex numbers, so you cannot speak of a sequence increasing or decreasing to a limit the way you can on the real line.