# 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

u_1 + u_2 + u_3 + . . .

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

s_1, s_2, s_3, . . .

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

s_n = u_1 + u_2 +
u_3 + . . . + 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:

## 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
z_1 + z_2 + z_3 + . . .

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

x_1 + x_2 + x_3 + . . .

and

y_1 + y_2 + y_3 + . . .

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
z_1 + z_2 + z_3 + . . .

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

|z_1| + |z_2| + |z_3| + . . .

converges, then the given series of complex
numbers converges.

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