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 for all n and x in a set E, and if the series ∑Mn converges, then the series ∑fn(x) converges uniformly on E
To apply the M-test, find a sequence of positive numbers {Mn} such that ∣fn(x)∣≤Mn for all n and x in the given set, and then check if the series ∑Mn converges
Example: For the series ∑n2xn on the interval [0,1], we can choose Mn=n21 since ∣n2xn∣≤n21 for all x∈[0,1] and ∑n21 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 ∑nxn 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 ∑nxn 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≤fn(x)≤gn(x) for all n and x in a set E, and if ∑gn(x) converges uniformly on E, then ∑fn(x) converges uniformly on E
Example: To show that ∑n3xn converges uniformly on [0,1], we can compare it with ∑n31, which converges
Conversely, if 0≤gn(x)≤fn(x) for all n and x in a set E, and if ∑gn(x) diverges, then ∑fn(x) diverges
Example: The series ∑nxn diverges on [1,∞) since it can be compared to the divergent harmonic series ∑n1
Ratio and Root Tests
The ratio test examines the limit of the ratio of consecutive terms in a series to determine convergence
If limn→∞supx∈E∣fn(x)fn+1(x)∣<1, then the series ∑fn(x) converges uniformly on E
Example: For the series ∑n!xn, the ratio test yields limn→∞∣(n+1)!xn+1⋅xnn!∣=limn→∞∣n+1x∣=0 for all x, so the series converges uniformly on R
The root test examines the limit of the nth root of the absolute value of the nth term in a series to determine convergence
If limn→∞supx∈E(∣fn(x)∣)1/n<1, then the series ∑fn(x) converges uniformly on E
Example: For the series ∑2nxn, the root test yields limn→∞(2n∣x∣n)1/n=2∣x∣, 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 ∑an(x−c)n, where c is the center of the series and an 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 limn→∞∣anan+1∣
If the limit exists and equals L, then R=L1. If L=0, then R=∞, and if L=∞, then R=0
Example: For the power series ∑nxn, the ratio test yields limn→∞∣1/n1/(n+1)∣=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 ∑nxn 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 ∑n2+1xn on [0,1], the M-test can be applied with Mn=n2+11 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(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 ∑nsin(nx) converges uniformly on 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 ∑n2+x2xn 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