Theory

Theory 1

A power series looks like this:

$$f(x)=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+\cdots$$

Power series are used to build and study functions. They allow a uniform “modeling framework” in which many functions can be described and compared. Power series are also convenient for computers because they provide a way to store and evaluate differentiable functions with numerical (approximate) values.

Small $x$ needed for power series

The most important fact about power series is that they work for small values of $x$.

Many power series diverge for $|x|$ too big; but even when they converge, for big $|x|$ they converge more slowly, and partial sum approximations are less accurate.

The idea of a power series is a modification of the idea of a geometric series in which the common ratio $r$ becomes a variable $x$, and each term has an additional coefficient parameter $a_n$ controlling the relative contribution of different orders.

Theory 2

Every power series has a radius of convergence and an interval of convergence.

Radius of convergence

Consider a power series centered at $x=0$:

$$f(x)=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+\cdots$$

Apply the ratio test:

$$\underset{n\to \infty }{\lim }\left|\frac{a_{n+1}{x}^{n+1}}{a_n{x}^n}\right|\gg \gg (\underset{n\to \infty }{\lim }\left|\frac{a_{n+1}}{a_n}\right|)|x|=L$$

Define the radius of convergence $R\in [0,\infty ]$:

$$R=\frac{1}{{\lim }_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|}$$

Therefore:

$$\begin{array}{cc}|x|<& R⟹L<1⟹\text{converges}\\ & \\ |x|>& R⟹L>1⟹\text{diverges}\end{array}$$

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We can build shifted power series for $x$ near some other value $c$. Just replace the variable $x$ with a shifted variable $u=x-c$:

$$\begin{array}{c}{a}_0+a_{1}u+a_2{u}^2+a_3{u}^3+\cdots \\ \\ \gg \gg a_0+a_1(x-c)+a_2{(x-c)}^2+a_3{(x-c)}^3+\cdots \end{array}$$

Now apply the ratio test to determine convergence:

$$\underset{n\to \infty }{\lim }\left|\frac{a_{n+1}|x-c{|}^{n+1}}{a_n|x-c{|}^n}\right|\gg \gg (\underset{n\to \infty }{\lim }\left|\frac{a_{n+1}}{a_n}\right|)|x-c|$$

Define the radius of convergence $R\in [0,\infty ]$:

$$R=\frac{1}{{\lim }_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|}$$

In the shifted setting, the radius of convergence limits the *distance from *:

$$\begin{array}{cc}|x-c|<& R⟹\text{converges}\\ & \\ |x-c|>& R⟹\text{diverges}\end{array}$$

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Method: To calculate the interval of convergence of a power series, follow these steps:

• Observe the center $c$ of the shifted series (or $c=0$ for no shift).
• Compute $R$ using the limit of coefficient ratios.
• Write down the preliminary interval $(c-R,c+R)$.
• Plug each endpoint, $c-R$ and $c-R$, into the original series
  • Check for convergence
• Add in the convergent endpoints. (4 possible scenarios.)