Introduction to Power Series

By the end of this lesson, you will be able to identify power series, evaluate them at specific values, recognize the geometric series as a power series, and understand convergence conceptually.

The Big Idea

A power series is an infinite series where each term is a power of $x$:

$$\sum_{n=0}^{\infty} a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots$$

Power series represent functions as infinite polynomials — the idea that a smooth function can be captured by an infinite sum of simpler terms. This is the gateway to Taylor series in calculus.

The Geometric Power Series

The infinite geometric series is the simplest power series:

$$\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \cdots \quad \text{for } |x| < 1$$

This converges when $|x| < 1$ (the ratio $x$ must be between $-1$ and $1$).

Radius and Interval of Convergence

Each power series converges for some values of $x$ and diverges for others.

Radius of convergence $R$: the series converges for $|x| < R$ and diverges for $|x| > R$.

Interval of convergence: the set of all $x$ for which the series converges (may include endpoints).

For the geometric series $\sum x^n$: $R = 1$, interval $(-1, 1)$.

Famous Power Series (Calculus Preview)

These are derived by Taylor's theorem in Calculus:

$$e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \quad (\text{all } x)$$

$$\sin x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{6} + \frac{x^5}{120} - \cdots$$

$$\cos x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots$$

$$\ln(1+x) = \sum_{n=1}^{\infty}\frac{(-1)^{n+1}x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots \quad (-1 < x \leq 1)$$

Example 1 — Evaluating a Power Series

$\displaystyle\sum_{n=0}^{\infty}\left(\frac{1}{2}\right)^n = \frac{1}{1-1/2} = 2$ (geometric, $|r| < 1$ ✓)

$\displaystyle\sum_{n=0}^{\infty} 2^n$: $|r| = 2 > 1$ → diverges.

Example 2 — Approximating $e$

$$e = \sum_{n=0}^{\infty}\frac{1}{n!} = 1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \cdots \approx 2.71828\ldots$$

After just 6 terms: $1 + 1 + 0.5 + 0.1\overline{6} + 0.04167 + 0.00833 = 2.7166$ (accurate to 3 decimal places).