Power Series Formula

Power series are an infinite series of the form _n=0^ a_n(x-c)^n = a_0 + a_1(x-c) + a_2(x-c)^2 + x s where c is the center and a_n are the coefficients.

The Formula

∑n=0∞an(x−c)n\sum_{n=0}^{\infty} a_n(x-c)^n
Radius of convergence: R=1lim sup⁡n→∞∣an∣1/nR = \frac{1}{\limsup_{n \to \infty} |a_n|^{1/n}} or use the ratio test: R=lim⁡n→∞∣anan+1∣R = \lim_{n \to \infty} \left|\frac{a_n}{a_{n+1}}\right|.

When to use: A power series is an 'infinite polynomial' centered at cc. For each value of xx, you get a number series that may or may not converge. The set of xx-values where it converges forms an interval centered at cc, and within that interval, the power series behaves like a well-defined function.

Quick Example

∑n=0∞xn=1+x+x2+x3+⋯=11−xfor ∣x∣<1\sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \cdots = \frac{1}{1-x} \quad \text{for } |x| < 1
At x=0.5x = 0.5: 1+0.5+0.25+0.125+⋯=2=11−0.51 + 0.5 + 0.25 + 0.125 + \cdots = 2 = \frac{1}{1-0.5}. ✓
At x=2x = 2: 1+2+4+8+⋯1 + 2 + 4 + 8 + \cdots diverges. ✗

Notation

RR = radius of convergence. Interval of convergence = (c−R,c+R)(c-R, c+R), with endpoints checked separately.

What This Formula Means

An infinite series of the form ∑n=0∞an(x−c)n=a0+a1(x−c)+a2(x−c)2+⋯\sum_{n=0}^{\infty} a_n(x-c)^n = a_0 + a_1(x-c) + a_2(x-c)^2 + \cdots where cc is the center and ana_n are the coefficients. A power series defines a function of xx wherever it converges.

A power series is an 'infinite polynomial' centered at cc. For each value of xx, you get a number series that may or may not converge. The set of xx-values where it converges forms an interval centered at cc, and within that interval, the power series behaves like a well-defined function.

Formal View

∑n=0∞an(x−c)n\sum_{n=0}^{\infty} a_n(x-c)^n has radius of convergence R=1/lim sup⁡n→∞∣an∣1/nR = 1 / \limsup_{n \to \infty} |a_n|^{1/n}. The series converges absolutely for ∣x−c∣<R|x - c| < R and diverges for ∣x−c∣>R|x - c| > R. Within (c−R,c+R)(c - R, c + R): ddx∑an(x−c)n=∑nan(x−c)n−1\frac{d}{dx}\sum a_n(x-c)^n = \sum n a_n(x-c)^{n-1} and ∫∑an(x−c)n dx=∑ann+1(x−c)n+1+C\int \sum a_n(x-c)^n\,dx = \sum \frac{a_n}{n+1}(x-c)^{n+1} + C.

Worked Examples

Example 1

medium
Find the radius of convergence of ∑n=0∞xnn+1\displaystyle\sum_{n=0}^{\infty} \frac{x^n}{n+1}.

Answer

R=1R = 1

First step

1
Apply the ratio test: compute ∣an+1an∣\left|\frac{a_{n+1}}{a_n}\right| where an=xnn+1a_n = \frac{x^n}{n+1}.

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Example 2

hard
Find the interval of convergence of ∑n=1∞(−1)nxnn\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n x^n}{n}.

Example 3

medium
Find the interval of convergence of ∑n=1∞(x−2)nn⋅3n\sum_{n=1}^{\infty} \frac{(x-2)^n}{n \cdot 3^n}.

Common Mistakes

  • Reporting only the radius and skipping endpoints - test x=c−Rx=c-R and x=c+Rx=c+R separately to get the full interval.
  • Confusing the center - powers are of (x−c)(x-c), so the interval is centered at cc, not at 0 unless c=0c=0.
  • Assuming it equals a function everywhere - a power series only defines a function inside its interval of convergence.

Why This Formula Matters

Power series let you manipulate functions term-by-term — differentiate, integrate, and combine them — and they are the home of Taylor series, generating functions, and many DE solutions. The central task is finding WHERE it converges (radius and interval), because outside it the 'function' doesn't exist. Recognizing it by "Is this a series whose terms are coefficients times powers of (x−c)(x-c), with convergence depending on the value of xx?" — rather than by familiar numbers — is what lets a student tell it apart from taylor series and polynomial and numeric (constant) series in a mixed problem set.

Frequently Asked Questions

What is the Power Series formula?

An infinite series of the form ∑n=0∞an(x−c)n=a0+a1(x−c)+a2(x−c)2+⋯\sum_{n=0}^{\infty} a_n(x-c)^n = a_0 + a_1(x-c) + a_2(x-c)^2 + \cdots where cc is the center and ana_n are the coefficients. A power series defines a function of xx wherever it converges.

How do you use the Power Series formula?

A power series is an 'infinite polynomial' centered at cc. For each value of xx, you get a number series that may or may not converge. The set of xx-values where it converges forms an interval centered at cc, and within that interval, the power series behaves like a well-defined function.

What do the symbols mean in the Power Series formula?

RR = radius of convergence. Interval of convergence = (c−R,c+R)(c-R, c+R), with endpoints checked separately.

Why is the Power Series formula important in Math?

Power series let you manipulate functions term-by-term — differentiate, integrate, and combine them — and they are the home of Taylor series, generating functions, and many DE solutions. The central task is finding WHERE it converges (radius and interval), because outside it the 'function' doesn't exist. Recognizing it by "Is this a series whose terms are coefficients times powers of (x−c)(x-c), with convergence depending on the value of xx?" — rather than by familiar numbers — is what lets a student tell it apart from taylor series and polynomial and numeric (constant) series in a mixed problem set.

What do students get wrong about Power Series?

The procedure for power series is the easy part; the trap is reporting only the radius and skipping endpoints. Asking "Is this a series whose terms are coefficients times powers of (x−c)(x-c), with convergence depending on the value of xx?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Power Series formula?

Before studying the Power Series formula, you should understand: convergence divergence, taylor series, sigma notation.

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