Intro to Mathematical Analysis Unit 13 ReviewPower Series and Taylor Expansions

Key Concepts and Definitions

• Power series: infinite series of the form $\sum_{n=0}^{\infty} a_n(x-c)^n$, where $a_n$ are coefficients, $x$ is a variable, and $c$ is the center of the series
• Radius of convergence: the radius $R$ of the largest open interval $(c-R, c+R)$ on which the power series converges
  • Inside the interval of convergence, the power series converges to a function $f(x)$
  • Outside the interval of convergence, the power series diverges
• Interval of convergence: the open interval $(c-R, c+R)$ where the power series converges
• Taylor series: a power series representation of a function $f(x)$ centered at a point $c$, given by $\sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n$
  • Maclaurin series: a special case of Taylor series centered at $c=0$
• Taylor polynomial: a finite sum of the first $n+1$ terms of a Taylor series, denoted by $P_n(x)=\sum_{k=0}^{n} \frac{f^{(k)}(c)}{k!}(x-c)^k$
• Remainder term: the difference between a function $f(x)$ and its Taylor polynomial $P_n(x)$, given by $R_n(x)=f(x)-P_n(x)$

Power Series Fundamentals

• Power series are infinite sums of terms involving powers of a variable $(x-c)$ and coefficients $a_n$
• The center $c$ of a power series is the point around which the series is expanded (often $c=0$)
• Convergence of a power series depends on the values of $x$ and the behavior of the coefficients $a_n$
• If a power series converges at a point $x=x_0$, it converges absolutely for all $x$ with $|x-c|<|x_0-c|$
• If a power series diverges at a point $x=x_1$, it diverges for all $x$ with $|x-c|>|x_1-c|$
• Power series can be used to represent functions and solve differential equations
• The sum of a power series, when it converges, is a function that can be differentiated or integrated term by term within its interval of convergence

Convergence and Radius of Convergence

• The radius of convergence $R$ determines the interval on which a power series converges
• To find the radius of convergence, use the ratio test: $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = L$
  • If $L < 1$, the series converges for all $x$ (infinite radius of convergence)
  • If $L > 1$, the series diverges for all $x \neq c$ (zero radius of convergence)
  • If $L = 1$, the ratio test is inconclusive, and further investigation is needed
• When $L = 1$, use the root test: $\lim_{n \to \infty} \sqrt[n]{|a_n|} = \rho$
  • If $\rho < 1$, the series converges for all $x$
  • If $\rho > 1$, the series diverges for all $x \neq c$
• The interval of convergence is $(c-R, c+R)$, where $R$ is the radius of convergence
• At the endpoints $x = c \pm R$, the series may converge conditionally, diverge, or converge absolutely (test using other methods)

Operations on Power Series

• Power series can be added, subtracted, multiplied, and divided (with some restrictions)
• Addition and subtraction: if two power series have the same center $c$ and converge on a common interval, they can be added or subtracted term by term
• Multiplication: if two power series converge on intervals $I_1$ and $I_2$, their product converges on the intersection of these intervals
  • The coefficients of the product series are given by the Cauchy product formula: $c_n = \sum_{k=0}^{n} a_k b_{n-k}$
• Division: if a power series $\sum_{n=0}^{\infty} a_n(x-c)^n$ converges to a function $f(x)$ and $a_0 \neq 0$, then there exists a power series $\sum_{n=0}^{\infty} b_n(x-c)^n$ that converges to $1/f(x)$ near $c$
  • The coefficients $b_n$ can be found using long division of power series
• Differentiation and integration: a power series can be differentiated or integrated term by term within its interval of convergence
  • The resulting series have the same radius of convergence as the original series

Taylor Series Basics

• Taylor series represent functions as power series centered at a point $c$
• The coefficients of a Taylor series are determined by the function's derivatives at the center $c$
• The Taylor series of a function $f(x)$ centered at $c$ is given by $\sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n$
  • If $c=0$, the series is called a Maclaurin series
• To find the Taylor series of a function, calculate its derivatives at the center and substitute them into the general formula
• Many common functions have known Taylor series expansions (exponential, trigonometric, logarithmic)
• Taylor series can be used to approximate functions, solve differential equations, and evaluate limits
• The interval of convergence of a Taylor series depends on the function and the center $c$

Taylor's Theorem and Approximations

• Taylor's theorem provides a way to approximate functions using Taylor polynomials and estimate the error
• The Taylor polynomial $P_n(x)$ of degree $n$ for a function $f(x)$ centered at $c$ is given by $P_n(x)=\sum_{k=0}^{n} \frac{f^{(k)}(c)}{k!}(x-c)^k$
• Taylor's theorem states that if $f(x)$ has $n+1$ continuous derivatives on an interval containing $c$, then $f(x) = P_n(x) + R_n(x)$
  • The remainder term $R_n(x)$ satisfies $\lim_{n \to \infty} R_n(x) = 0$ for all $x$ in the interval of convergence
• Lagrange's form of the remainder: $R_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x-c)^{n+1}$, where $\xi$ is between $c$ and $x$
• Taylor polynomials can be used to approximate functions near the center $c$
• The accuracy of the approximation increases with the degree $n$ of the polynomial
• The error in the approximation is bounded by the magnitude of the remainder term $R_n(x)$

Applications and Examples

• Power series and Taylor series have numerous applications in mathematics, physics, and engineering
• Approximating functions: Taylor polynomials can be used to approximate functions near a point
  • Example: $\sin(x) \approx x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}$ for $x$ near $0$
• Solving differential equations: power series can be used to find solutions to linear differential equations
  • Example: $y'' + y = 0$ has a solution $y(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}x^{2n}$
• Evaluating limits: L'Hôpital's rule can be applied to limits involving Taylor series
  • Example: $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$ can be evaluated using the Maclaurin series of $\sin(x)$
• Modeling physical phenomena: many laws in physics and engineering can be expressed using power series
  • Example: the displacement of a vibrating string can be modeled using a Fourier series (a type of power series)

Common Pitfalls and Tips

• When finding the radius of convergence, be careful with the ratio test when the limit is 1 (use the root test or other methods)
• Remember that the ratio test and the root test only determine the radius of convergence, not the convergence at the endpoints
• When using Taylor's theorem, ensure that the function has the required number of continuous derivatives on the interval
• Be aware that a Taylor series may converge to a function on a smaller interval than the function's domain
• When approximating functions using Taylor polynomials, consider the trade-off between accuracy and computational complexity
• Remember that power series can be differentiated or integrated term by term only within their interval of convergence
• When solving differential equations using power series, make sure to check that the solution satisfies the initial or boundary conditions
• Practice manipulating power series and identifying their intervals of convergence to develop a strong understanding of their properties