Welcome back, class! Tonight we represent a major shift in our study of Calculus. Up until now, we have been looking at series of numbers. In Section 11.8, we transition to Power Series, which you can think of as polynomials with an infinite number of terms.

What is a Power Series?

A power series is a series of the form:

$$ \sum_{n=0}^{\infty} c_n x^n = c_0 + c_1x + c_2x^2 + c_3x^3 + \dots $$

More generally, a series centered at $a$ is written as:

$$ \sum_{n=0}^{\infty} c_n (x-a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + \dots $$

Here, $x$ is a variable, and the $c_n$ terms are coefficients. The convergence of the series depends on the value of $x$.

Key Concepts: Radius and Interval of Convergence

For any given power series, there are exactly three possibilities regarding its convergence:

  1. The series converges only when $x = a$ (the center). In this case, the Radius of Convergence $R = 0$.
  2. The series converges for all real numbers $x$. In this case, $R = \infty$.
  3. The series converges for $|x-a| < R$ and diverges for $|x-a| > R$. This creates an Interval of Convergence between $(a-R, a+R)$.

The Strategy: Using the Ratio Test

To find the Radius ($R$) and Interval ($I$) of convergence, we almost always rely on the Ratio Test. Here is the step-by-step process we cover in the video and notes:

  • Step 1: Set up the Limit. Calculate $ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| $.
  • Step 2: Solve the Inequality. Set the result less than 1 (since the Ratio Test requires $L < 1$ for convergence) and solve for $|x-a|$. This will identify your radius $R$.
  • Step 3: Check the Endpoints! This is the most common place for errors. The Ratio Test is inconclusive when $L=1$, which happens exactly at the endpoints of your interval. You must plug these $x$ values back into the original series and test them individually (often using the Alternating Series Test or P-Series Test).

By the end of this lesson, you will be comfortable determining exactly where these infinite polynomials are valid. Be sure to follow along with the attached notes as we work through several examples involving factorials and geometric patterns.

Happy Studying!