Power Series Formula

The Formula

\sum_{n=0}^{\infty} a_n(x-c)^n
Radius of convergence: R = \frac{1}{\limsup_{n \to \infty} |a_n|^{1/n}} or use the ratio test: 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 c. For each value of x, you get a number series that may or may not converge. The set of x-values where it converges forms an interval centered at c, and within that interval, the power series behaves like a well-defined function.

Quick Example

\sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \cdots = \frac{1}{1-x} \quad \text{for } |x| < 1
At x = 0.5: 1 + 0.5 + 0.25 + 0.125 + \cdots = 2 = \frac{1}{1-0.5}. ✓
At x = 2: 1 + 2 + 4 + 8 + \cdots diverges. ✗

Notation

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

What This Formula Means

An infinite series of the form \sum_{n=0}^{\infty} a_n(x-c)^n = a_0 + a_1(x-c) + a_2(x-c)^2 + \cdots where c is the center and a_n are the coefficients. A power series defines a function of x wherever it converges.

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

Formal View

\sum_{n=0}^{\infty} a_n(x-c)^n has radius of convergence R = 1 / \limsup_{n \to \infty} |a_n|^{1/n}. The series converges absolutely for |x - c| < R and diverges for |x - c| > R. Within (c - R, c + R): \frac{d}{dx}\sum a_n(x-c)^n = \sum n a_n(x-c)^{n-1} and \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 \displaystyle\sum_{n=0}^{\infty} \frac{x^n}{n+1}.

Solution

  1. 1
    Apply the ratio test: compute \left|\frac{a_{n+1}}{a_n}\right| where a_n = \frac{x^n}{n+1}.
  2. 2
    Simplify the ratio: \left|\frac{x^{n+1}}{n+2} \cdot \frac{n+1}{x^n}\right| = |x|\cdot\frac{n+1}{n+2} \xrightarrow{n\to\infty} |x|
  3. 3
    The series converges when L = |x| < 1, so the radius of convergence is R = 1.

Answer

R = 1
The ratio test extracts R by finding the limiting ratio. Here L = |x|, so R=1.

Example 2

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

Common Mistakes

  • Forgetting to check the endpoints of the interval of convergence: the ratio/root test is inconclusive at x = c \pm R, so you must substitute these values and test each one separately.
  • Assuming the radius of convergence is infinite: \sum \frac{x^n}{n!} converges for all x (R = \infty), but \sum n! \, x^n converges only at x = 0 (R = 0). Most power series have finite, nonzero R.
  • Differentiating or integrating without adjusting the radius: term-by-term differentiation and integration preserve the radius of convergence (but may change endpoint behavior).

Why This Formula Matters

Power series are the backbone of mathematical analysis. Taylor series are a special case. Power series represent solutions to DEs (Bessel functions, Airy functions), define special functions, and enable term-by-term operations that simplify complex calculations.

Frequently Asked Questions

What is the Power Series formula?

An infinite series of the form \sum_{n=0}^{\infty} a_n(x-c)^n = a_0 + a_1(x-c) + a_2(x-c)^2 + \cdots where c is the center and a_n are the coefficients. A power series defines a function of x wherever it converges.

How do you use the Power Series formula?

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

What do the symbols mean in the Power Series formula?

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

Why is the Power Series formula important in Math?

Power series are the backbone of mathematical analysis. Taylor series are a special case. Power series represent solutions to DEs (Bessel functions, Airy functions), define special functions, and enable term-by-term operations that simplify complex calculations.

What do students get wrong about Power Series?

Finding the radius of convergence is usually straightforward (ratio or root test), but checking the endpoints requires separate analysis—often using alternating series test or p-series comparison. Don't forget the endpoints!

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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