Power series/R/Introduction/Section

Definition

Let ${\left(c_n\right)}_{n\in \mathbb{N}}$ be a sequence of real numbers and $x$ another real number. Then the series

$\sum _{n=0}^{\mathrm{\infty }}{c}_n{x}^n$

is called the power series in $x$ for the coefficients ${\left(c_n\right)}_{n\in \mathbb{N}}$.

For a power series, it is important that $x$ varies and that the power series represents in some convergence interval a function in $x$. Every polynomial is a power series, but one for which all coefficients starting with a certain member are $0$. In this case, the convergence is everywhere.

We have encountered an important power series earlier, the geometric series $\sum _{n=0}^{\mathrm{\infty }}{x}^n$ (here all coefficients equal $1$), which converges for $|x|<1$ and represents the function $1/(1-x)$, 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.

Theorem

Let

$f(x):=\sum _{n=0}^{\mathrm{\infty }}{c}_n{x}^n$

be a power series and suppose that there exists some $x_0\ne 0$ such that $\sum _{n=0}^{\mathrm{\infty }}{c}_n{x}_0^n$ converges. Then there exists a positive $R$ (where $R=\mathrm{\infty }$ is allowed) such that for all $x\in ℝ$ fulfilling $|x|<R$ the series converges absolutely. On such an (open) interval of convergence, the power series $f(x)$ represents a

continuous function.

Proof

The proof needs a systematic study of power series and of limits of sequences of functions. We will not do this here.

$◻$

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.

Definition

For two series $\sum _{i=0}^{\mathrm{\infty }}{a}_i$ and $\sum _{j=0}^{\mathrm{\infty }}{b}_j$ of real numbers, the series

$\sum _{k=0}^{\mathrm{\infty }}{c}_k\text{ with }{c}_k:=\sum _{i=0}^k{a}_i{b}_{k-i}$

is called the Cauchy-product of the series.

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

Lemma

Let

$\sum _{k=0}^{\mathrm{\infty }}{a}_k\text{ and }\sum _{k=0}^{\mathrm{\infty }}{b}_k$

be absolutely convergent series of real numbers. Then also the Cauchy product $\sum _{k=0}^{\mathrm{\infty }}{c}_k$ is absolutely convergent and for its sum the equation

$\sum _{k=0}^{\mathrm{\infty }}{c}_k=\left(\sum _{k=0}^{\mathrm{\infty }}{a}_k\right)\cdot \left(\sum _{k=0}^{\mathrm{\infty }}{b}_k\right)$

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.