Such series are often used to define functions, such as

Obviously, if we are going to define new functions in this
way, it is important to know when such a series converges.
As indicated in the last equation, we generally think of
the numbers _{
} as fixed numbers (or
fixed coefficients) and the

Power series are also used to define functions on the complex numbers too:

for

The main result about power series is that each power series
has a radius of convergence,

This result, i.e., the existence of such a radius of convergence,
in fact works equally well for the complexes as for the reals.
The terminology radius

arises from thinking about the set
of complex numbers

The radius of convergence, *diverges* for all
possible values of _{
}
_{0}.)

One further point to bear in mind is that the theorem on
radius of convergence says nothing about what actually happens on
the perimeter of this circular disc, i.e., when

We shall consider a power series
_{
}

Theorem.

**Proof.**

Surprisingly, perhaps, this does all the work we need, and to get the radius of convergence theorem we just need to combine the last theorem with an old result about completeness of the reals.

Theorem on Radius of Convergence.

For all
power series _{
}

**Proof.**

**Subproof.**

**Subproof.**

Therefore _{
}

On the other hand,

**Subproof.**

In this case, is a bounded nonempty set
(nonempty because it contains 0) and so by the
supremum form of the completeness of reals
there is a least upper bound _{
}

**Subproof.**

So _{
converges absolutely
}

**Subproof.**

So _{
does not converge
}

Example.

The series ^{
}

**Proof.**

The ratio of consecutive terms of ^{
}
^{
+1
}
^{
}

On the radius of convergence
we have ^{
}
^{
}

Example.

The series ^{
}

**Proof.**

The ratio of consecutive terms of
^{
}
^{-1}

Example.

The series ^{
}
^{2}

**Proof.**

Again, the ratio of consecutive terms of
^{
}
^{2
}
^{2}
^{-1+
-2
}
^{-2}

Example.

The series ^{
}

**Proof.**

The ratio of consecutive terms of
^{
}

Example.

The series ^{
}

**Proof.**

For ^{
}
^{-1}
^{2}
^{2-1}
^{2-2}
^{2}
^{
2
}
^{
}
^{2}
^{-1}
^{
}
^{2}
^{
}

Hence from the null sequence test that the series
^{
}
^{
}
^{
}