power series

a power series centered at $a$ is defined with $c_n\in \mathbb{R}$, whereby:

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

meaning it is written as $c_0+c_1(x-a)+c_2(x-a{)}^2+c_3(x-a{)}^3+\cdots$

radius of convergence

• there is a radius of convergence $R\ge 0$ for any power series, possibly infinite, by which the series is absolutely convergent where $|x-a|<R$, and it does not converge when $|x-a|>R$ , the case where $|x-a|=R$ is uncertain
• ratio test: if all coefficients $c_n$ are nonzero, and some $\underset{n\to \mathrm{\infty }}{lim}\left|\frac{c_n}{c_{n+1}}\right|$ evaluates to some $c$ — if $c$ is positive or $+\mathrm{\infty }$, then that limit is equivalent to the radius of convergence
• Taylor’s Formula: a power series $f(x)$ can be differentiated, integrated on the bounds of $(a-R,a+R)$, the derivatives and integrals will have radius of convergence $R$ and $c_n=\frac{f^{(n)}(a)}{n!}$ to construct the series

linear combinations of power series

When $\sum _{n=0}^{\mathrm{\infty }}{a}_n$ and $\sum _{n=0}^{\mathrm{\infty }}{b}_n$ are both convergent, linear combinations of them can be described in the usual fashion:

$$c_1\sum _{n=0}^{\mathrm{\infty }}{a}_n+c_2\sum _{n=0}^{\mathrm{\infty }}{b}_n=\sum _{n=0}^{\mathrm{\infty }}{c}_1{a}_n+c_2{b}_n$$

some power series

geometric series

$$1+r+r^2+r^3+\cdots =\sum _{n=0}^{\mathrm{\infty }}{r}^n=\frac{1}{1-r}$$

which converges $-1<r<1$, and diverges otherwise.

exponential series

$$1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots =\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}=e^x$$

which converges for all $x\in \mathbb{R}$.

absolutely convergent

If:

$$\sum _{n=0}^{\mathrm{\infty }}|a_n|$$

converges, then:

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

also converges.

This situation is called absolutely convergent.