Chapter 8: Infinite Sequences and Series
Section 8.4: Power Series
Essentials
Many problems in applied mathematics and mathematical physics are solved by an infinite sum of functions, rather than by just a sum of numbers. Such series have names like Fourier series, Fourier-Bessel series, Fourier-Legendre series, etc. The intent of this present section is to consider series of functions where the functions are powers of x, or powers of x−c for some fixed constant c. These are called power series for the obvious reason.
Definition 8.4.1: Power Series
Infinite series of the form ∑n=0∞an xn are called power series, and the constants an are called the coefficients.
A series of the form ∑n=0∞anx−cn is called a power series centered at c. The behavior of such power series is exactly parallel to that of the series centered at c=0.
The convergence of a power series is determined pointwise: the variable x is replaced throughout with a constant x0 and the convergence of the resulting series of numbers, namely Σ an x0n, is determined as per Section 8.3. If this series of numbers converges, then the series is said to converge for x=x0. The totality of x-values for which the power series converges forms the domain of a function fx whose rule is the power series.
Table 8.4.1 lists the salient facts about the power series Σ anxn that converges to define a function fx.
The domain of f consists of at least the point x=0.
Convergence takes place in an interval whose form is one of the following:
−R,R,−R,R,−R,R,−R,r
The interval is called the interval of convergence.
The number R is called the radius of convergence; R could be zero, a finite number greater than zero, or infinity. If R=∞, then the interval of convergence is −R,R.
The radius of convergence of Σ anxn is given by R=limn→∞anan+1, whenever this limit exists; similarly, R=1/limn→∞ann, again provided this limit exists. These two formulas are based respectively on the Ratio and Root tests, the second one using the conventions 1/0⇒R=∞ and 1/∞⇒R=0. (Hadamard's definitive formula for R is beyond the scope of a first calculus course.)
The power series converges absolutely inside the interval of convergence.
Convergence of Σ anxn at x= ±R must be tested individually. The endpoint behavior could be divergence, conditional convergence, or absolute convergence. However, if the series converges absolutely at one endpoint, it will necessarily converge absolutely at the other.
Inside the interval of convergence, the power series can be differentiated termwise so that f′x=Σ n anxn−1.
Inside the interval of convergence, all derivatives of f exist, and an=fn0/n! Consequently, if Σ anx−cn converges to fx, then in the interval of convergence fx=∑fncn!x−cn.
Table 8.4.1 Properties of power series
The repetition of Σ anxn in item (5) in the table is deliberate. The radius of convergence for this series can be established by the Ratio test as follows. The series converges if, by the Ratio test, limn→∞an+1 xn+1an xn<1. For fixed x, routine algebra gives x<limn→∞|anan+1|≡R.
Consider instead, the series Σ an x2 n and apply the same reasoning. The Ratio test now gives
limn→∞an+1 x2n+1an x2 n<1 and x2 <limn→∞|anan+1| so that x <limn→∞|anan+1|≡R
In other words, the recipe for R is predicated on the form of the powers in the power series.
The radius of convergence for a power series in powers of α x+β must take into account the factor α. Indeed, the radius of convergence will then be 1αlimn→∞|anan+1| because the calculations that lead to R in the simpler case will lead to α x+β<limn→∞an/an+1. If the limit on the right is taken as λ, the inequality then implies x+β/α<λ/α. The term β/α shifts the center of the interval of convergence to x=−β/α, but the factor 1/α scales λ. The endpoints of the interval of convergence are then x=−β/α−λ/α and x=−β/α+λ/α.
Examples
For each power series listed in Table 8.4.2, determine the radius of convergence and the interval of convergence. Even though (7) in Table 8.4.1 claims that absolute convergence at one end of the interval of convergence implies absolute convergence at the other, in each case where convergence at an endpoint is absolute, verify that it also absolute at the other.
Each sum in Table 8.4.2 is to infinity, but that symbol has been deleted to save vertical space.
Example 8.4.1
∑n=1nn xn2nn+1!
Example 8.4.13
∑n=1n 2 x+3n
Example 8.4.2
∑n=0xnn+3
Example 8.4.14
∑n=2xnn lnn
Example 8.4.3
∑n=1−1n+1xnn5
Example 8.4.15
∑n=0x2 n2 n!
Example 8.4.4
∑n=1n xn
Example 8.4.16
∑n=1−1n+1n5 x−3n
Example 8.4.5
∑n=1xnn2
Example 8.4.17
∑n=12nnn3 x+5n
Example 8.4.6
∑n=0xnn!
Example 8.4.18
∑n=02nn!3 x+5n
Example 8.4.7
∑n=1nn xn
Example 8.4.19
∑n=17 x−3nn n+1
Example 8.4.8
∑n=1xnn 3n
Example 8.4.20
∑n=1n28nx3 n
Example 8.4.9
∑n=1n 3n xn
Example 8.4.21
∑n=2n!2 xn2 n!
Example 8.4.10
∑n=15n xnn+32
Example 8.4.22
∑k=0−1k x2 k+nk! k+n! 22 k+n
Example 8.4.11
∑n=1n32n xn
Example 8.4.23
∑n=1n22n4 x+5n
Example 8.4.12
∑n=2xnlnn
Example 8.4.24
∑n=11⋅3⋅⋯⋅2 n−12⋅5⋅⋯⋅3 n−1xn
Table 8.4.2 Radius of convergence and interval of convergence to be determined for each power series
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