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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

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  • The Weierstrass M-test provides a sufficient condition for the uniform convergence of a series of functions on a set
  • States that if fn(x)Mn|f_n(x)| \leq M_n for all nn and xx in a set EE, and if the series Mn\sum M_n converges, then the series fn(x)\sum f_n(x) converges uniformly on EE
  • To apply the M-test, find a sequence of positive numbers {Mn}\{M_n\} such that fn(x)Mn|f_n(x)| \leq M_n for all nn and xx in the given set, and then check if the series Mn\sum M_n converges
    • Example: For the series xnn2\sum \frac{x^n}{n^2} on the interval [0,1][0, 1], we can choose Mn=1n2M_n = \frac{1}{n^2} since xnn21n2|\frac{x^n}{n^2}| \leq \frac{1}{n^2} for all x[0,1]x \in [0, 1] and 1n2\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 xnn\sum \frac{x^n}{n} converges uniformly on [0,r][0, r] for any r<1r < 1, so it also converges pointwise on [0,r][0, r]
  • However, the converse is not always true a series may converge pointwise on a set without converging uniformly
    • Example: The series xnn\sum \frac{x^n}{n} converges pointwise on [0,1][0, 1] but does not converge uniformly on [0,1][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 0fn(x)gn(x)0 \leq f_n(x) \leq g_n(x) for all nn and xx in a set EE, and if gn(x)\sum g_n(x) converges uniformly on EE, then fn(x)\sum f_n(x) converges uniformly on EE
    • Example: To show that xnn3\sum \frac{x^n}{n^3} converges uniformly on [0,1][0, 1], we can compare it with 1n3\sum \frac{1}{n^3}, which converges
  • Conversely, if 0gn(x)fn(x)0 \leq g_n(x) \leq f_n(x) for all nn and xx in a set EE, and if gn(x)\sum g_n(x) diverges, then fn(x)\sum f_n(x) diverges
    • Example: The series xnn\sum \frac{x^n}{n} diverges on [1,)[1, \infty) since it can be compared to the divergent harmonic series 1n\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 limnsupxEfn+1(x)fn(x)<1\lim_{n\to\infty} \sup_{x\in E} |\frac{f_{n+1}(x)}{f_n(x)}| < 1, then the series fn(x)\sum f_n(x) converges uniformly on EE
    • Example: For the series xnn!\sum \frac{x^n}{n!}, the ratio test yields limnxn+1(n+1)!n!xn=limnxn+1=0\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 xx, so the series converges uniformly on R\mathbb{R}
  • The root test examines the limit of the nnth root of the absolute value of the nnth term in a series to determine convergence
  • If limnsupxE(fn(x))1/n<1\lim_{n\to\infty} \sup_{x\in E} (|f_n(x)|)^{1/n} < 1, then the series fn(x)\sum f_n(x) converges uniformly on EE
    • Example: For the series xn2n\sum \frac{x^n}{2^n}, the root test yields limn(xn2n)1/n=x2\lim_{n\to\infty} (\frac{|x|^n}{2^n})^{1/n} = \frac{|x|}{2}, so the series converges uniformly on any interval [r,r][-r, r] with r<2r < 2

Radius and Interval of Convergence

Power Series and Radius of Convergence

  • A power series is a series of the form an(xc)n\sum a_n(x-c)^n, where cc is the center of the series and ana_n are the coefficients
  • The radius of convergence RR is the radius of the largest open interval centered at cc on which the power series converges
  • The ratio test is commonly used to find the radius of convergence by evaluating limnan+1an\lim_{n\to\infty} |\frac{a_{n+1}}{a_n}|
    • If the limit exists and equals LL, then R=1LR = \frac{1}{L}. If L=0L = 0, then R=R = \infty, and if L=L = \infty, then R=0R = 0
    • Example: For the power series xnn\sum \frac{x^n}{n}, the ratio test yields limn1/(n+1)1/n=1\lim_{n\to\infty} |\frac{1/(n+1)}{1/n}| = 1, so the radius of convergence is R=1R = 1

Interval of Convergence

  • The interval of convergence is the open interval (cR,c+R)(c-R, c+R), with possible convergence at the endpoints determined by testing the series at x=cRx = c-R and x=c+Rx = c+R
  • To find the interval of convergence, first determine the radius of convergence RR, then test the series at the endpoints cRc-R and c+Rc+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 xnn\sum \frac{x^n}{n} centered at c=0c = 0, the radius of convergence is R=1R = 1. Testing the endpoints, the series converges at x=1x = -1 but diverges at x=1x = 1, so the interval of convergence is [1,1)[-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 xnn2+1\sum \frac{x^n}{n^2+1} on [0,1][0, 1], the M-test can be applied with Mn=1n2+1M_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 n(x2)n\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 sin(nx)n\sum \frac{\sin(nx)}{n} converges uniformly on R\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 xnn2+x2\sum \frac{x^n}{n^2+x^2} on [0,1][0, 1], first use the M-test to show uniform convergence on [0,r][0, r] for any r<1r < 1, then use the comparison test to show pointwise convergence at x=1x = 1


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© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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