➗Calculus II Unit 6 – Power Series

What are Power Series?

• Power series are infinite series where each term is a constant multiplied by a variable raised to a non-negative integer power
• General form of a power series: $\sum_{n=0}^{\infty} a_n(x-c)^n = a_0 + a_1(x-c) + a_2(x-c)^2 + \ldots$
  • $a_n$ represents the coefficients
  • $c$ is the center of the series
  • $x$ is the variable
• Power series can be used to represent functions as an infinite sum of terms
• The coefficients of a power series can be determined by various methods such as using the definition of the function or using Taylor series
• Power series allow for the approximation of functions near a specific point (the center of the series)
• The behavior of a power series depends on the values of $x$ and the convergence of the series
• Power series have a radius and interval of convergence that determine where the series converges or diverges

Convergence and Divergence

• Convergence of a power series means that the infinite sum approaches a finite value as the number of terms approaches infinity
• Divergence of a power series means that the infinite sum does not approach a finite value or tends to infinity as the number of terms approaches infinity
• The convergence or divergence of a power series depends on the values of $x$ and the coefficients $a_n$
• The ratio test can be used to determine the convergence or divergence of a power series
  • If $\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| < 1$, the series converges
  • If $\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| > 1$, the series diverges
  • If $\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = 1$, the test is inconclusive
• The root test can also be used to determine convergence or divergence
  • If $\lim_{n \to \infty} \sqrt[n]{|a_n|} < 1$, the series converges
  • If $\lim_{n \to \infty} \sqrt[n]{|a_n|} > 1$, the series diverges
• Absolute convergence implies convergence, but a series may converge conditionally without absolute convergence

Radius and Interval of Convergence

• The radius of convergence $R$ is the range of values for $x$ where the power series converges
• The interval of convergence is the set of $x$ values for which the series converges
  • It can be written as $(c-R, c+R)$, where $c$ is the center of the series
• To find the radius of convergence, use the ratio test or the root test
  • Ratio test: $R = \lim_{n \to \infty} \left|\frac{a_n}{a_{n+1}}\right|$
  • Root test: $R = \frac{1}{\lim_{n \to \infty} \sqrt[n]{|a_n|}}$
• If $R = 0$, the series converges only at the center $c$
• If $R = \infty$, the series converges for all $x$
• To find the interval of convergence, find the radius of convergence and then check the endpoints $(c-R)$ and $(c+R)$ for convergence
• The behavior of the series at the endpoints needs to be checked separately using other tests (such as the alternating series test or the comparison test)

Operations on Power Series

• Power series can be added, subtracted, multiplied, and divided under certain conditions
• Addition and subtraction of power series:
  • Add or subtract the coefficients of like terms
  • The resulting series has the same radius of convergence as the original series
• Multiplication of power series:
  • Multiply the series term by term and combine like terms
  • The radius of convergence of the product is at least the smaller of the radii of convergence of the two original series
• Division of power series:
  • Divide the series term by term using long division
  • The radius of convergence of the quotient is at least the smaller of the radii of convergence of the two original series
• Differentiation of power series:
  • Differentiate the series term by term
  • The resulting series has the same radius of convergence as the original series
• Integration of power series:
  • Integrate the series term by term
  • The resulting series has the same radius of convergence as the original series

Taylor and Maclaurin Series

• Taylor series are power series representations of functions centered at a specific point $x=a$
• The Taylor series of a function $f(x)$ centered at $x=a$ is given by:
  • $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$
  • $f^{(n)}(a)$ represents the $n$-th derivative of $f$ evaluated at $x=a$
• Maclaurin series are a special case of Taylor series centered at $x=0$
• The Maclaurin series of a function $f(x)$ is given by:
  • $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$
• To find the Taylor or Maclaurin series of a function:
  1. Find the derivatives of the function at the center point
  2. Evaluate the derivatives at the center point
  3. Substitute the values into the Taylor or Maclaurin series formula
• Taylor and Maclaurin series can be used to approximate functions near the center point
• The accuracy of the approximation depends on the number of terms used and the proximity to the center point

Applications of Power Series

• Power series have numerous applications in mathematics, science, and engineering
• Approximating functions:
  • Power series can be used to approximate complicated functions near a specific point
  • This is useful when the original function is difficult to evaluate or integrate
• Solving differential equations:
  • Some differential equations can be solved using power series methods
  • The solution is expressed as a power series, and the coefficients are determined by substituting the series into the differential equation
• Modeling physical phenomena:
  • Many physical systems can be modeled using power series
  • Examples include the motion of a pendulum, the vibration of a string, and the flow of heat in a material
• Calculating integrals and derivatives:
  • Power series can be integrated or differentiated term by term, providing a way to calculate integrals and derivatives of functions that are difficult to evaluate directly
• Approximating special functions:
  • Special functions, such as the exponential function, sine, and cosine, can be represented using their Maclaurin series
  • These series approximations are used in numerical computations and analysis

Common Power Series Examples

• Geometric series: $\sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \ldots$
  • Converges for $|x| < 1$
  • Sum of the series: $\frac{1}{1-x}$
• Exponential series: $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots$
  • Converges for all $x$
• Sine series: $\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots$
  • Converges for all $x$
• Cosine series: $\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}x^{2n} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots$
  • Converges for all $x$
• Logarithmic series: $\ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}x^n = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots$
  • Converges for $|x| < 1$
• Binomial series: $(1+x)^{\alpha} = \sum_{n=0}^{\infty} \binom{\alpha}{n}x^n = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2!}x^2 + \frac{\alpha(\alpha-1)(\alpha-2)}{3!}x^3 + \ldots$
  • Converges for $|x| < 1$ and all $\alpha$, or for $|x| \leq 1$ and $\alpha > 0$

Tips and Tricks for Working with Power Series

• When given a power series, identify the center $c$ and the coefficients $a_n$
• Use the ratio test or the root test to find the radius of convergence
• Check the endpoints of the interval of convergence separately
• When adding or subtracting power series, make sure they have the same center and variable
• When multiplying or dividing power series, the resulting series has a radius of convergence at least as large as the smaller of the radii of the original series
• To find the Taylor series of a function, use the formula and evaluate the derivatives at the center point
• When using power series to approximate functions, consider the number of terms needed for the desired accuracy
• Look for opportunities to represent functions using known power series (e.g., exponential, sine, cosine)
• When solving differential equations using power series, substitute the series into the equation and solve for the coefficients
• Practice manipulating and working with power series to develop familiarity and intuition