11.3 Tests for Convergence of Series

Tests for convergence of series are crucial tools in mathematical analysis. They help us determine whether infinite sums of functions converge uniformly or pointwise on specific intervals. Understanding these tests is key to analyzing the behavior of series and their properties.

The Weierstrass M-test, comparison test, ratio test, and root test are essential for determining uniform convergence. For power series, finding the radius and interval of convergence is crucial. These techniques allow us to analyze complex series and their convergence properties effectively.

Weierstrass M-test for Uniform Convergence

Sufficient Condition for Uniform Convergence

• The Weierstrass M-test provides a sufficient condition for the uniform convergence of a series of functions on a set
• States that if $|f_n(x)| \leq M_n$ for all $n$ and $x$ in a set $E$, and if the series $\sum M_n$ converges, then the series $\sum f_n(x)$ converges uniformly on $E$
• To apply the M-test, find a sequence of positive numbers $\{M_n\}$ such that $|f_n(x)| \leq M_n$ for all $n$ and $x$ in the given set, and then check if the series $\sum M_n$ converges
  • Example: For the series $\sum \frac{x^n}{n^2}$ on the interval $[0, 1]$, we can choose $M_n = \frac{1}{n^2}$ since $|\frac{x^n}{n^2}| \leq \frac{1}{n^2}$ for all $x \in [0, 1]$ and $\sum \frac{1}{n^2}$ converges

Relationship to Pointwise Convergence

• If the M-test conditions are satisfied, the series of functions converges uniformly on the given set
• Uniform convergence implies pointwise convergence, meaning that if a series converges uniformly on a set, it also converges pointwise on that set
  • Example: The series $\sum \frac{x^n}{n}$ converges uniformly on $[0, r]$ for any $r < 1$, so it also converges pointwise on $[0, r]$
• However, the converse is not always true a series may converge pointwise on a set without converging uniformly
  • Example: The series $\sum \frac{x^n}{n}$ converges pointwise on $[0, 1]$ but does not converge uniformly on $[0, 1]$

Convergence Tests for Series of Functions

Comparison Test

• The comparison test compares a given series with a known convergent or divergent series to determine its convergence
• If $0 \leq f_n(x) \leq g_n(x)$ for all $n$ and $x$ in a set $E$, and if $\sum g_n(x)$ converges uniformly on $E$, then $\sum f_n(x)$ converges uniformly on $E$
  • Example: To show that $\sum \frac{x^n}{n^3}$ converges uniformly on $[0, 1]$, we can compare it with $\sum \frac{1}{n^3}$, which converges
• Conversely, if $0 \leq g_n(x) \leq f_n(x)$ for all $n$ and $x$ in a set $E$, and if $\sum g_n(x)$ diverges, then $\sum f_n(x)$ diverges
  • Example: The series $\sum \frac{x^n}{n}$ diverges on $[1, \infty)$ since it can be compared to the divergent harmonic series $\sum \frac{1}{n}$

Ratio and Root Tests

• The ratio test examines the limit of the ratio of consecutive terms in a series to determine convergence
• If $\lim_{n\to\infty} \sup_{x\in E} |\frac{f_{n+1}(x)}{f_n(x)}| < 1$, then the series $\sum f_n(x)$ converges uniformly on $E$
  • Example: For the series $\sum \frac{x^n}{n!}$, the ratio test yields $\lim_{n\to\infty} |\frac{x^{n+1}}{(n+1)!} \cdot \frac{n!}{x^n}| = \lim_{n\to\infty} |\frac{x}{n+1}| = 0$ for all $x$, so the series converges uniformly on $\mathbb{R}$
• The root test examines the limit of the $n$th root of the absolute value of the $n$th term in a series to determine convergence
• If $\lim_{n\to\infty} \sup_{x\in E} (|f_n(x)|)^{1/n} < 1$, then the series $\sum f_n(x)$ converges uniformly on $E$
  • Example: For the series $\sum \frac{x^n}{2^n}$, the root test yields $\lim_{n\to\infty} (\frac{|x|^n}{2^n})^{1/n} = \frac{|x|}{2}$, so the series converges uniformly on any interval $[-r, r]$ with $r < 2$

Radius and Interval of Convergence

Power Series and Radius of Convergence

• A power series is a series of the form $\sum a_n(x-c)^n$, 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
• The ratio test is commonly used to find the radius of convergence by evaluating $\lim_{n\to\infty} |\frac{a_{n+1}}{a_n}|$
  • If the limit exists and equals $L$, then $R = \frac{1}{L}$. If $L = 0$, then $R = \infty$, and if $L = \infty$, then $R = 0$
  • Example: For the power series $\sum \frac{x^n}{n}$, the ratio test yields $\lim_{n\to\infty} |\frac{1/(n+1)}{1/n}| = 1$, so the radius of convergence is $R = 1$

Interval of Convergence

• The interval of convergence is the open interval $(c-R, c+R)$, with possible convergence at the endpoints determined by testing the series at $x = c-R$ and $x = c+R$
• To find the interval of convergence, first determine the radius of convergence $R$, then test the series at the endpoints $c-R$ and $c+R$
  • If the series converges at an endpoint, include that endpoint in the interval of convergence
  • If the series diverges at an endpoint, exclude that endpoint from the interval of convergence
  • Example: For the power series $\sum \frac{x^n}{n}$ centered at $c = 0$, the radius of convergence is $R = 1$. Testing the endpoints, the series converges at $x = -1$ but diverges at $x = 1$, so the interval of convergence is $[-1, 1)$

Convergence Analysis of Series of Functions

Choosing Appropriate Tests and Techniques

• Different tests and techniques can be used to analyze the convergence of series of functions, depending on the properties of the series
• The Weierstrass M-test, comparison test, ratio test, and root test are common tools for determining uniform convergence of series of functions
  • Example: For the series $\sum \frac{x^n}{n^2+1}$ on $[0, 1]$, the M-test can be applied with $M_n = \frac{1}{n^2+1}$ to show uniform convergence
• For power series, finding the radius and interval of convergence is a key step in analyzing their convergence behavior
  • Example: To analyze the convergence of the power series $\sum n(x-2)^n$, first find the radius of convergence using the ratio test, then determine the interval of convergence by testing the endpoints

Combining Multiple Tests and Techniques

• Other techniques, such as the integral test or Dirichlet's test, may be applicable in specific cases
  • Example: The integral test can be used to show that the series $\sum \frac{\sin(nx)}{n}$ converges uniformly on $\mathbb{R}$
• It is essential to choose the most appropriate test or technique based on the characteristics of the given series of functions
• Combining multiple tests or techniques may be necessary to fully analyze the convergence of a series of functions
  • Example: To analyze the series $\sum \frac{x^n}{n^2+x^2}$ on $[0, 1]$, first use the M-test to show uniform convergence on $[0, r]$ for any $r < 1$, then use the comparison test to show pointwise convergence at $x = 1$