# Power series/R/Introduction/Section

Let be a sequence of real numbers and another real number. Then the series

*power series*in for the coefficients .

For a power series, it is important that varies and that the power series represents in some *convergence interval* a function in . Every polynomial is a power series, but one for which all coefficients starting with a certain member are . In this case, the convergence is everywhere.

We have encountered an important power series earlier, the geometric series (here all coefficients equal ), which converges for
and represents the function , see
fact.
Another important power series is the *exponential series*, which for every real number converges and represents the *real exponential function*. Its inverse function is the *natural logarithm*.

The behavior of convergence of a power series is given by the following theorem.

Let

be a power series and suppose that there exists some such that converges. Then there exists a positive (where is allowed) such that for all fulfilling the series converges absolutely. On such an (open) interval of convergence, the power series represents a continuous function.

### Proof

If two functions are given by power series, then their sum is simply given by the
(componentwise defined)
sum of the power series. It is not clear at all by which power series the product of two power series is described. The answer is given by the Cauchy-product of series.

Also, for the following statement we do not provide a proof.

Let

be absolutely convergent series of real numbers. Then also the Cauchy product is absolutely convergent and for its sum the equation

holds.

From this we can infer that the product of power series is given by the power series whose coefficients are those which arise by the multiplication of polynomials, see
exercise.