Convergent Infinite Series

The textbook definition of convergence

For the most part we shall be satisfied with the discussion of convergent infinite series in the text.

The basic idea is that if

u₁ + u₂ + u₃ + . . .

is an infinite series, one forms the sequence of partial sums

s₁, s₂, s₃, . . .

where the n^th partial sum s_n is defined by the formula

s_n = u₁ + u₂ + u₃ + . . . + u_n.

The series is (definition) said to converge with sum s if s is the limit of the sequence of partial sums, that is, if

s = lim_{{n = infinity}} s_n .

Extending the textbook definition to complex series

We wish to allow infinite series of complex numbers. That is, in the preceding discussion we wish to allow the terms u_j of the infinite series to be complex numbers.

The partial sums s_n are formed in the same way.

As above, the sum is the limit of the sequence of partial sums.

But what does it mean to take the limit of a sequence of complex numbers?

The notion of limit of a sequence of complex numbers rests on understanding:

what is meant by the distance d(z₁, z₂) between two complex numbers z₁ and z₂.
that the sequence of distances d(s, s_n) is a sequence of real numbers.
that for complex numbers (definition):
s = lim_{{n = infinity}} s_n
if and only if
lim_{{n = infinity}} d(s_n, s) = 0 .

The distance between two complex numbers

The complex number z = x + i y may be represented geometrically by the point (x, y) in the Cartesian plane.

The distance between two complex numbers is the distance between the two corresponding points in the Cartesian plane.

Summing a Complex Series is Equivalent to Summing 2 Real Series

The summing by convergence of an infinite series of complex numbers is equivalent to the summing by convergence of two real infinite series, one consisting of the real parts of the given series and the other consisting of the imaginary parts. That is, if z_n = x_n + i y_n, where x_n and y_n are real, then the infinite series

z₁ + z₂ + z₃ + . . .

converges with sum z if and only if the two real infinite series

x₁ + x₂ + x₃ + . . .

and

y₁ + y₂ + y₃ + . . .

are both convergent, and if x is the sum of the first and y the sum of the second, then z = x + i y.

Absolute Convergence Imples Convergence for Complex Series

For infinite series of complex numbers, as with infinite series of real numbers, absolute convergence implies convergence. That is, if

z₁ + z₂ + z₃ + . . .

is a series of complex numbers for which the series of absolute values

|z₁| + |z₂| + |z₃| + . . .

converges, then the given series of complex numbers converges.

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