Power series are infinite sums of terms with increasing powers. They're crucial for representing functions, solving equations, and approximating values. You'll learn how to construct, manipulate, and analyze these series.

Understanding power series convergence is key. You'll explore methods to determine where a series converges and how to represent functions as power series. This knowledge opens doors to advanced mathematical techniques and applications.

Power Series

Construction of power series

• Definition: a power series is an infinite series of the form $\sum_{n=0}^{\infty} a_n(x-c)^n$ where $a_n$ are coefficients, $c$ is the center, and $x$ is the variable
  • Maclaurin series: a special case of power series centered at $c=0$ (e.g., $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ for $e^x$)
  • Taylor series: power series centered at any point $c$ (e.g., $\sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n$ general form)
• Generating power series from known series involves substituting expressions for the variable
  • Given the Maclaurin series for $e^x$, find the power series for $e^{x^2}$ by substituting $x^2$ for $x$: $\sum_{n=0}^{\infty} \frac{(x^2)^n}{n!}$

Radius of convergence calculation

• The radius of convergence determines the range of values for which the power series converges
  • Ratio test: $\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| = L$
    • $L < 1$: series converges for $|x-c| < \frac{1}{L}$
    • $L > 1$: series diverges for $|x-c| > \frac{1}{L}$
    • $L = 1$: further investigation needed (root test, comparison test, etc.)
  • For the series $\sum_{n=1}^{\infty} \frac{x^n}{n}$, the radius of convergence is 1
• The interval of convergence is the set of $x$ values for which the series converges
  • Determined by the radius of convergence and behavior at the endpoints ($x=c \pm R$)
  • For the series $\sum_{n=1}^{\infty} \frac{x^n}{n}$, the interval of convergence is $(-1, 1]$

Functions as power series representations

• Representing functions as power series
  • Taylor series expansion: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n$ (e.g., expand $f(x) = \ln(x)$ as a Taylor series centered at $c=1$)
  • Maclaurin series expansion: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$ (e.g., expand $f(x) = \cos(x)$ as a Maclaurin series)
• Power series representations are significant for
  • Approximating functions: estimate function values near the center of the series (e.g., approximate $\sin(0.1)$ using the Maclaurin series for $\sin(x)$)
  • Solving differential equations: power series can solve certain types of differential equations (e.g., solve $y'' - xy = 0$ using power series)
  • Defining new functions: power series can define functions that cannot be expressed in closed form (e.g., the series $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}x^{2n+1}$ defines $\sin(x)$)

Functions as Power Series

Functions as power series representations

• Representing functions as power series
  • Binomial series: $(1+x)^r = \sum_{n=0}^{\infty} \binom{r}{n}x^n$ for $|x|<1$ (e.g., expand $(1+x)^{-1/2}$ as a power series)
  • Geometric series: $\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$ for $|x|<1$ (e.g., expand $\frac{1}{1+x^2}$ as a power series)
  • Exponential series: $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$ for all $x$ (e.g., expand $e^{\sin(x)}$ as a power series)
• Operations on power series
  • Addition/subtraction: add/subtract coefficients of like terms (e.g., find the power series for $\cos(x) + \sin(x)$)
  • Multiplication: multiply series term by term and collect like terms (e.g., find the power series for $\sin(x)\cos(x)$)
  • Differentiation: differentiate term by term (e.g., find the power series for $\frac{d}{dx}e^x$)
  • Integration: integrate term by term (e.g., find the power series for $\int_0^x \sin(t) dt$)
• Power series have various applications
  • Evaluating limits: power series can evaluate limits that are difficult to compute directly (e.g., evaluate $\lim_{x\to 0} \frac{e^x-1}{x}$ using power series)
  • Approximating integrals: power series can approximate definite integrals (e.g., approximate $\int_0^1 \frac{1}{1+x^2} dx$ using power series)

Convergence and Divergence of Power Series

• Convergence of a power series occurs when the sum of its terms approaches a finite limit
  • The radius of convergence determines the range of x-values for which the series converges
  • Absolute convergence: when the series of absolute values of the terms converges
• Divergence of a power series happens when the sum of its terms does not approach a finite limit
  • A series diverges outside its radius of convergence
  • Conditional convergence: when a series converges, but not absolutely

Analytic Functions and Laurent Series

• An analytic function is infinitely differentiable and can be represented by its Taylor series
  • All power series define analytic functions within their radius of convergence
  • Analytic functions have unique power series representations
• Laurent series are generalizations of power series that include negative powers
  • They can represent functions with singularities
  • Laurent series take the form $\sum_{n=-\infty}^{\infty} a_n(x-c)^n$
  • Used to study behavior of functions near singularities and in complex analysis