Power Series

Power Series

Definition 1. A power series about $x=a$ is a series of the form $$\sum_{n=0}^{\infty} c_n(x-a)^n$$ in which the center $a$ and the coefficients $c_0, c_1, ..., c_n$ are constants.

Convergent Theorem for Power Series

Theorem 1. If the power series $\sum_{n=0}^{\infty} a_n x^n$ converges at $x = c \neq 0$, then it converges absolutely for all $x$ with $|x| < |c|$. If the series diverges at $x=d$, then it diverges for all $x$ with $|x| > |d|$.

Corollary. The convergence of the series $\sum c_n(x-a)^n$ is described by one of the following three cases:
(1) There is a positive number $R$ such that the series diverges for $x$ with $|x - a| > R$ but converges absolutely for $x$ with $|x - a| < R$. The series may or may not converge at either of the endpoints $x = a - R$ and $x = a + R$.
(2) The series converges absolutely for every $x (R = \infty)$.
(3) The series converges at $x = a$ and diverges elsewhere $(R = 0)$.

Radius of Convergence

Definition 2. The number $R$ shown in the Corollary is called the radius of convergence of the power series, and the interval of radius $R$ centered at $x=a$ is called the interval of convergence. 

Series Multiplication for Power Series

Theorem 2. If $A(x) = \sum_{n=0}^{\infty} a_nx^n$ and $B(x) = \sum_{n=0}^{\infty} b_nx^n$ converge absolutely for $|x| < R$, and $$c_n = a_0b_n + a_1b_{n-1} + \cdots + a_{n-1}b_1 + a_nb_0 = \sum_{k=0}^{n} a_kb_{n-k},$$ then $\sum_{n=0}^{\infty} c_nx^n$ converges absolutely to $A(x)B(x)$ for $|x| < R$; $$\left( \sum_{n=0}^{\infty} a_nx^n \right) \left( \sum_{n=0}^{\infty} b_nx^n \right) = \sum_{n=0}^{\infty} c_nx^n.$$

Substitution Rule for Power Series

Theorem 3. If $\sum_{n=0}^{\infty} a_nx^n$ converges absolutely for $|x| < R$ and $f$ is continuous function, then $\sum_{n=0}^{\infty} a_n(f(x))^n$ converges absolutely on the set of points $x$ where $|f(x)| < R$.

Differentiation Rule for Power Series

Theorem 4. If $\sum c_n(x-a)^n$ has radius of convergence $R > 0$, it defines a function $$f(x) = \sum_{n=0}^{\infty} c_n(x-a)^n \text{  on the interval  } a-R < x < a+R.$$ This function $f$ has derivatives of all orders inside the interval, and we obtain the derivatives by differentiating the original series term by term: $$f'(x) = \sum_{n=1}^{\infty} nc_n (x-a)^{n-1}, \\ f''(x) = \sum_{n=2}^{\infty} n(n-1)c_n(x-a)^{n-2},$$ and so on. Each of these derived series converges at every point of the interval $a-R < x < a+R$.

Integration Rule for Power Series

Theorem 5. Suppose that $$f(x) = \sum_{n=0}^{\infty} c_n(x-a)^n$$ converges for $a-R < x < a+R (R > 0)$. Then $$\sum_{n=0}^{\infty} c_n \frac{(x-a)^{n+1}}{n+1}$$ converges for $a-R < x < a+R$ and $$\int f(x) dx = \sum_{n=0}^{\infty} c_n \frac{(x-a)^{n+1}}{n+1} + C$$ for $a - R < x < a+ R$.