Power Series

From this section, we move from series of constant terms to series of function terms.

The reason we introduce this kind of series is to learn how to use polynomials to approach a given function(Taylor Expansion). We now have learned how to test the common series(without functions or self-variable). The next step is to expand it into series of function terms.

The Definition of Power Series

A series in the form:

${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{C}_n(x-a{)}^n=C_0+C_1(x-a)+C_2(x-a{)}^2+\dots }$ is called the power series.

e.g.

For the simplest power series ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{x}^n}$, we apply the absolute ratio test to it, the output is:

${\displaystyle \rho =|x|}$

So the series converges when ${\displaystyle |x|<1}$, diverges when ${\displaystyle |x|>1}$, and when ${\displaystyle |x|=1}$, the series equals to add many ${\displaystyle 1}$ together and it obviously diverges.

So ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{x}^n}$ converges when ${\displaystyle |x|<1}$, diverges when ${\displaystyle |x|\ge 1}$.

The Core Idea of D​dealing With Power Series

In The Test of Positive Series, we introduced two important questions when facing a infinite series. Now we also need to expand them.

For power series, we add a question:

• What is its convergence interval or convergence domain?

That is, for what interval of ${\displaystyle x}$ the series converges.

Abel's Theorem and The Radius of Convergence

Abel's Theorem

For a power series ${\displaystyle \sum a_n{x}^n}$:

• If it converges at ${\displaystyle x=c}$, then it converges for all ${\displaystyle |x|<|c|}$.
• If it diverges at ${\displaystyle x=d}$, then it diverges for all ${\displaystyle |x|>|d|}$.

We will prove it in the future, but not now.

The Radius of Convergence

From the Abel's Theorem, since the series converges when ${\displaystyle |x|<|c|}$, we say that when ${\displaystyle x}$ in ${\displaystyle (-c,c)}$, the series converges, and ${\displaystyle c}$ is the radius of convergence.

When handling with a power series, we can always consider the convergent radius exists. That is because the definition of the radius of convergence includes all possible real numbers ${\displaystyle x}$.

So when finding the convergent sets of a power series, all we nee to do is to follow the below steps:

1. Find ${\displaystyle \rho =\underset{n\to \mathrm{\infty }}{lim}|\frac{a_{n+1}}{a_n}|}$.
2. Let ${\displaystyle R=\frac{1}{\rho }}$, and that is the convergent radius.
3. Determine if the series converges or diverges at the endpoints.
4. Merge the intervals and determine the absolutely convergent intervals and divergent intervals.

Then we can try to find the sum function of the power series.

Operations on Power Series