12.2 Power series

Power series are infinite sums that can represent functions as polynomials. They're a key tool in calculus, allowing us to approximate complex functions and analyze their behavior over different intervals.

In this part of the chapter, we'll learn how to work with power series, find their intervals of convergence, and use them to represent functions. We'll also explore techniques for differentiating and integrating power series term by term.

Power series and convergence

Definition and components

• A power series is an infinite series of the form $\sum_{n=0}^{\infty} a_n(x-c)^n$, where:
  • $a_n$ are the coefficients
  • $c$ is the center
  • $x$ is the variable
• The radius of convergence, $R$, is the distance from the center $c$ to the nearest point where the series diverges or fails to be defined
  • It can be computed using the ratio test: $R = \lim_{n\to\infty} \left|\frac{a_n}{a_{n+1}}\right|$

Convergence behavior

• Within the interval of convergence $(c-R, c+R)$, the power series converges
  • Outside this interval, it diverges
  • At the endpoints, the series may converge or diverge
• If $R = \infty$, the series converges for all $x$
  • Example: $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ (Maclaurin series for $e^x$)
• If $R = 0$, the series converges only at $x = c$
  • Example: $\sum_{n=0}^{\infty} n!x^n$ (centered at $c=0$)
• The interval of convergence includes the endpoints if the series converges at those points

Interval of convergence

Finding the interval

• To find the interval of convergence, first compute the radius of convergence $R$ using the ratio test
• If $R$ is finite, check the endpoints $c-R$ and $c+R$ by substituting them into the series to determine convergence or divergence
  • If the series converges at both endpoints, the interval of convergence is $[c-R, c+R]$
  • If it diverges at both endpoints, the interval is $(c-R, c+R)$
  • If the series converges at one endpoint and diverges at the other, the interval is either $[c-R, c+R)$ or $(c-R, c+R]$, depending on which endpoint is included

Examples

• For the series $\sum_{n=0}^{\infty} \frac{x^n}{n+1}$, the ratio test gives $R = 1$
  • Checking the endpoints $x=-1$ and $x=1$ reveals that the series converges at both
  • Therefore, the interval of convergence is $[-1, 1]$
• For the series $\sum_{n=1}^{\infty} \frac{(x-2)^n}{n}$, the ratio test gives $R = 1$
  • Checking the endpoints $x=1$ and $x=3$ reveals that the series diverges at both (harmonic series)
  • Therefore, the interval of convergence is $(1, 3)$

Term-by-term operations

Differentiation

• Power series can be differentiated term by term within their interval of convergence
• To differentiate a power series $\sum_{n=0}^{\infty} a_n(x-c)^n$:
  1. Multiply each term by $n$
  2. Decrease the exponent by 1
  • Resulting series: $\sum_{n=1}^{\infty} na_n(x-c)^{n-1}$
• The resulting series after differentiation has the same interval of convergence as the original series

Integration

• Power series can be integrated term by term within their interval of convergence
• To integrate a power series $\sum_{n=0}^{\infty} a_n(x-c)^n$:
  1. Divide each term by $n+1$
  2. Increase the exponent by 1
  • Resulting series: $\sum_{n=0}^{\infty} \frac{a_n}{n+1}(x-c)^{n+1} + C$
• The resulting series after integration has the same interval of convergence as the original series

Power series representations

Taylor and Maclaurin series

• Taylor series and Maclaurin series are power series representations of functions centered at a specific point
• The Taylor series of a function $f(x)$ centered at $x=c$ is given by: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n$
  • $f^{(n)}(c)$ denotes the $n$th derivative of $f$ evaluated at $c$
• The Maclaurin series is a special case of the Taylor series centered at $c=0$: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$

Finding Taylor and Maclaurin series

• To find the Taylor or Maclaurin series of a function:
  1. Compute the derivatives of the function at the center point
  2. Substitute them into the series formula
• The interval of convergence for a Taylor or Maclaurin series is determined by the ratio test, as with any power series

Common Maclaurin series expansions

• $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$
  • Interval of convergence: $(-\infty, \infty)$
• $\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}x^{2n+1}$
  • Interval of convergence: $(-\infty, \infty)$
• $\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}x^{2n}$
  • Interval of convergence: $(-\infty, \infty)$