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
Top images from around the web for Definition and properties
real analysis - Proof of the "Radius of Convergence Theorem" - Mathematics Stack Exchange View original
Is this image relevant?
complex analysis - to find radius of convergence of power series. - Mathematics Stack Exchange View original
Is this image relevant?
calculus - Interval of convergence of a power series, with a check for convergence at endpoints ... View original
Is this image relevant?
real analysis - Proof of the "Radius of Convergence Theorem" - Mathematics Stack Exchange View original
Is this image relevant?
complex analysis - to find radius of convergence of power series. - Mathematics Stack Exchange View original
Is this image relevant?
1 of 3
Top images from around the web for Definition and properties
real analysis - Proof of the "Radius of Convergence Theorem" - Mathematics Stack Exchange View original
Is this image relevant?
complex analysis - to find radius of convergence of power series. - Mathematics Stack Exchange View original
Is this image relevant?
calculus - Interval of convergence of a power series, with a check for convergence at endpoints ... View original
Is this image relevant?
real analysis - Proof of the "Radius of Convergence Theorem" - Mathematics Stack Exchange View original
Is this image relevant?
complex analysis - to find radius of convergence of power series. - Mathematics Stack Exchange View original
Is this image relevant?
1 of 3
A power series is a series of the form ∑n=0∞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
If a power series converges at a point x0, it converges absolutely for all x satisfying ∣x−c∣<∣x0−c∣
If a power series diverges at x0, it diverges for all x satisfying ∣x−c∣>∣x0−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:
∑n=0∞xn has an infinite radius of convergence
∑n=0∞n!xn 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 limn→∞∣an+1/an∣ exists, then the radius of convergence is R=1/limn→∞∣an+1/an∣
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 ∞, the radius of convergence is 0
Example: For ∑n=0∞n!xn, limn→∞∣anan+1∣=limn→∞∣n+11∣=0, so R=∞
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 ∑n=0∞2nxn, limn→∞∣anan+1∣=limn→∞∣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:
Determine the radius of convergence R using the ratio test
If R is infinite, the interval of convergence is (−∞,∞)
If R is zero, the interval of convergence is the single point {c}
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 ∑n=0∞n2xn, R=1. Testing endpoints:
At x=−1: ∑n=0∞n2(−1)n converges by the alternating series test
At x=1: ∑n=0∞n21 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 ∑n=0∞n(x−1)n, R=1
At x=0: ∑n=0∞n(−1)n converges conditionally by the alternating series test
At x=2: ∑n=0∞n1 diverges by the p-series test with p=1≤1