Power Series and Taylor Series: Definitions, Convergence, and Operations

Power Series

Definition of Power Series

A power series is an infinite series of the form:

• About x = 0: $\sum_{n=0}^{\infty} c_n x^n = c_0 + c_1 x + c_2 x^2 + \cdots + c_n x^n + \cdots$

• About x = a: $\sum_{n=0}^{\infty} c_n (x-a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + \cdots + c_n(x-a)^n + \cdots$

Here, a is the center of the series and the coefficients $c_0, c_1, c_2, \ldots$ are constants.

Convergence of Power Series

The convergence of a power series depends on the value of $x$:

• There exists a radius of convergence $R \geq 0$ such that the series converges absolutely for $|x-a| < R$ and diverges for $|x-a| > R$.

• At the endpoints $x = a \pm R$, convergence must be checked separately.

The Convergence Theorem for Power Series states:

• If $\sum_{n=0}^{\infty} a_n x^n$ 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 $|x| > |d|$.

Interval and Radius of Convergence

The interval of convergence is the set of all $x$ for which the series converges. The radius of convergence $R$ is half the length of this interval.

• For $|x-a| < R$, the series converges absolutely.

• For $|x-a| > R$, the series diverges.

• At $|x-a| = R$, convergence must be checked separately.

Operations with Power Series

Multiplication of Power Series

Power series can be multiplied like polynomials. If $A(x) = \sum_{n=0}^{\infty} a_n x^n$ and $B(x) = \sum_{n=0}^{\infty} b_n x^n$ converge absolutely for $|x| < R$, then their product is:

• $A(x)B(x) = \sum_{n=0}^{\infty} c_n x^n$, where $c_n = a_0 b_n + a_1 b_{n-1} + \cdots + a_n b_0 = \sum_{k=0}^n a_k b_{n-k}$

Substitution in Power Series

If $\sum_{n=0}^{\infty} a_n x^n$ converges absolutely for $|x| < R$ and $f$ is continuous, then $\sum_{n=0}^{\infty} a_n [f(x)]^n$ converges absolutely for $|f(x)| < R$.

Example: Power Series Expansion

To find the power series for $\frac{x^3}{x+2}$, multiply $x^3$ by the series for $\frac{1}{x+2}$:

• $\frac{x^3}{x+2} = x^3 \sum_{n=0}^{\infty} \frac{(-1)^n}{2^{n+1}} x^n = \sum_{n=0}^{\infty} \frac{(-1)^n}{2^{n+1}} x^{n+3}$

• First few terms: $\frac{1}{2}x^3 - \frac{1}{4}x^4 + \frac{1}{8}x^5 - \frac{1}{16}x^6 + \cdots$

Differentiation and Integration of Power Series

Term-by-Term Differentiation

If $\sum_{n=0}^{\infty} c_n (x-a)^n$ has radius of convergence $R > 0$, then it can be differentiated term by term within $|x-a| < R$:

• $f(x) = \sum_{n=0}^{\infty} c_n (x-a)^n$

• $f'(x) = \sum_{n=1}^{\infty} n c_n (x-a)^{n-1}$

• $f''(x) = \sum_{n=2}^{\infty} n(n-1) c_n (x-a)^{n-2}$

Term-by-Term Integration

Similarly, integration can be performed term by term:

• $\int f(x) dx = \sum_{n=0}^{\infty} c_n \frac{(x-a)^{n+1}}{n+1} + C$

Taylor and Maclaurin Series

Definition of Taylor and Maclaurin Series

The Taylor series of a function $f$ at $x = a$ is:

• $\sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!} (x-a)^k$

The Maclaurin series is the Taylor series at $a = 0$:

• $\sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!} x^k$

Taylor Polynomials

The Taylor polynomial of order n for $f$ at $x = a$ is:

• $P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n$

Example: Taylor Polynomials for $e^x$

• $P_1(x) = 1 + x$

• $P_2(x) = 1 + x + \frac{x^2}{2!}$

• $P_3(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}$

Frequently Used Taylor Series

Function	Taylor Series	Interval
$\frac{1}{1-x}$	$\sum_{n=0}^{\infty} x^n$	$|x| < 1$
$\frac{1}{1+x}$	$\sum_{n=0}^{\infty} (-1)^n x^n$	$|x| < 1$
$e^x$	$\sum_{n=0}^{\infty} \frac{x^n}{n!}$	$|x| < \infty$
$\sin x$	$\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}$	$|x| < \infty$
$\cos x$	$\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}$	$|x| < \infty$
$\ln(1+x)$	$\sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n}$	$-1 < x \leq 1$
$\tan^{-1} x$	$\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2n+1}$	$|x| \leq 1$