Introduction to Power Series: Learn It 1

  • Recognize power series and when they converge
  • Find where a power series converges and where it doesn’t
  • Use power series to write functions

What is a Power Series?

A power series is a special type of infinite series where each term contains a variable raised to increasing powers. Think of it as an “infinite polynomial” — instead of stopping at [latex]x^3[/latex] or [latex]x^{10}[/latex], the powers keep going forever.

The most basic power series looks like this:

[latex]\displaystyle\sum_{n=0}^{\infty} c_n x^n = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots[/latex]

Here, [latex]x[/latex] is our variable and the [latex]c_n[/latex] values are constants called coefficients.

You’ve actually seen a power series before. The geometric series [latex]1 + x + x^2 + x^3 + \cdots[/latex] is a power series where all coefficients equal [latex]1[/latex].

This series converges when [latex]|x| < 1[/latex] and diverges when [latex]|x| \geq 1[/latex]. This gives us our first hint that power series don’t converge everywhere — the value of [latex]x[/latex] matters.

power series centered at zero

A series of the form

[latex]\displaystyle\sum_{n=0}^{\infty} c_n x^n = c_0 + c_1 x + c_2 x^2 + \cdots[/latex]

is called a power series centered at [latex]x = 0[/latex].

power series centered at [latex]a[/latex]

A series of the form

[latex]\displaystyle\sum_{n=0}^{\infty} c_n (x-a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + \cdots[/latex]

is called a power series centered at [latex]x = a[/latex].

We always define [latex]x^0 = 1[/latex] and [latex]{(x-a)}^{0}=1[/latex] , even when [latex]x = 0[/latex] or [latex]x = a[/latex]. This ensures our series starts with the constant term [latex]c_0[/latex].

Here are several power series to help you recognize the pattern.

Centered at [latex]x = 0[/latex]:

  • [latex]\displaystyle\sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots[/latex]
  • [latex]\displaystyle\sum_{n=0}^{\infty} n! x^n = 1 + x + 2! x^2 + 3! x^3 + \cdots[/latex]

Centered at [latex]x = 2[/latex]:

  • [latex]\displaystyle\sum_{n=0}^{\infty} \frac{(x-2)^n}{(n+1) \cdot 3^n} = 1 + \frac{x-2}{2 \cdot 3} + \frac{(x-2)^2}{3 \cdot 3^2} + \frac{(x-2)^3}{4 \cdot 3^3} + \cdots[/latex]