13.2 Radius and Interval of Convergence

Power series are the mathematical equivalent of a Swiss Army knife. They're versatile tools that let us represent functions as infinite sums of terms. But how far can we stretch these series before they break?

That's where the radius and interval of convergence come in. They tell us the range of x-values where a power series behaves nicely, giving us a playground for manipulating and analyzing functions in new ways.

Radius of convergence for power series

Definition and properties

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• A power series is a series of the form $\sum_{n=0}^{\infty} a_n(x-c)^n$, where $c$ is the center of the series and $a_n$ are the coefficients
• The radius of convergence $R$ is the radius of the largest open interval centered at $c$ on which the power series converges
• If a power series converges at a point $x_0$, it converges absolutely for all $x$ satisfying $|x-c| < |x_0-c|$
• If a power series diverges at $x_0$, it diverges for all $x$ satisfying $|x-c| > |x_0-c|$

Types of convergence radii

• The radius of convergence can be finite, infinite, or zero
  • If $R$ is finite, the series converges absolutely for $|x-c| < R$ and diverges for $|x-c| > R$
  • If $R$ is infinite, the series converges for all $x$
  • If $R$ is zero, the series converges only at $x = c$
• Examples:
  • $\sum_{n=0}^{\infty} x^n$ has an infinite radius of convergence
  • $\sum_{n=0}^{\infty} n!x^n$ has a radius of convergence of 0

Ratio test for convergence

Applying the ratio test

• The ratio test determines the radius of convergence of a power series
  • If $\lim_{n\to\infty} |a_{n+1}/a_n|$ exists, then the radius of convergence is $R = 1/\lim_{n\to\infty} |a_{n+1}/a_n|$
• To apply the ratio test, find the limit of the absolute value of the ratio of successive coefficients
  • If the limit is $L$, then $R = 1/L$
  • If the limit is 0, the radius of convergence is infinite
  • If the limit is $\infty$, the radius of convergence is 0
• Example: For $\sum_{n=0}^{\infty} \frac{x^n}{n!}$, $\lim_{n\to\infty} |\frac{a_{n+1}}{a_n}| = \lim_{n\to\infty} |\frac{1}{n+1}| = 0$, so $R = \infty$

Interpreting the limit

• If the limit does not exist or is a finite non-zero value, the radius of convergence is the reciprocal of that value
• Example: For $\sum_{n=0}^{\infty} 2^n x^n$, $\lim_{n\to\infty} |\frac{a_{n+1}}{a_n}| = \lim_{n\to\infty} |2| = 2$, so $R = 1/2$

Interval of convergence

Finding the interval of convergence

• The interval of convergence is the set of all values of $x$ for which the power series converges, always centered at $c$
• To find the interval of convergence:
  1. Determine the radius of convergence $R$ using the ratio test
  2. If $R$ is infinite, the interval of convergence is $(-\infty, \infty)$
  3. If $R$ is zero, the interval of convergence is the single point $\{c\}$
  4. If $R$ is finite, the interval of convergence is at least $(c-R, c+R)$

Testing endpoints for convergence

• To determine if the endpoints are included in the interval of convergence, test the series for convergence at $x = c-R$ and $x = c+R$
• Use other convergence tests such as the alternating series test, p-series test, or comparison test
• Example: For $\sum_{n=0}^{\infty} \frac{x^n}{n^2}$, $R = 1$. Testing endpoints:
  • At $x = -1$: $\sum_{n=0}^{\infty} \frac{(-1)^n}{n^2}$ converges by the alternating series test
  • At $x = 1$: $\sum_{n=0}^{\infty} \frac{1}{n^2}$ converges by the p-series test with $p=2>1$
  • The interval of convergence is $[-1, 1]$

Power series behavior at endpoints

Convergence and divergence at endpoints

• The behavior of a power series at the endpoints of its interval of convergence depends on whether the series converges or diverges at those points
• If the series converges at an endpoint, that endpoint is included in the interval of convergence
  • The series may converge conditionally or absolutely at the endpoint
• If the series diverges at an endpoint, that endpoint is not included in the interval of convergence

Determining endpoint behavior

• To determine the behavior at the endpoints, substitute the endpoint values into the power series
• Use appropriate convergence tests to determine if the series converges or diverges
• The behavior at the endpoints can differ for the left and right endpoints
  • One endpoint may be included while the other is not, or both may be included or excluded
• Example: For $\sum_{n=0}^{\infty} \frac{(x-1)^n}{n}$, $R = 1$
  • At $x = 0$: $\sum_{n=0}^{\infty} \frac{(-1)^n}{n}$ converges conditionally by the alternating series test
  • At $x = 2$: $\sum_{n=0}^{\infty} \frac{1}{n}$ diverges by the p-series test with $p=1\leq 1$
  • The interval of convergence is $[0, 2)$