Section 11.8: Power Series

Welcome to Professor Baker's Math Class! Today, we will be diving into Sections 11.8 and 11.9, focusing on power series and how they can represent functions. Let's begin by understanding what a power series is. A power series is a series of the form:

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

Here, $x$ is a variable, and the $c_n$'s are constants known as the coefficients of the series. For each fixed $x$, the series is a series of constants that can be tested for convergence or divergence. A power series may converge for some values of $x$ and diverge for others. The sum of the series is a function, such as:

$$ f(x) = c_0 + c_1x + c_2x^2 + ... + c_nx^n + ... $$

The domain of this function, $f(x)$, is the set of all $x$ for which the series converges. Notice that $f$ resembles a polynomial, but it has infinitely many terms.

Key Concepts:

  • Convergence: A power series may converge for certain values of $x$.
  • Divergence: A power series may diverge for certain values of $x$.
  • Coefficients: The constants $c_n$ in the power series.

Example: Consider the geometric series:

$$ \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + ... $$

This series converges when $ -1 < x < 1 $ and diverges when $|x| \ge 1$.

More generally, we can define a power series centered at $a$ as:

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

Notice that when $x = a$, all terms are 0 for $n \ge 1$, and the power series always converges.

Important Theorem: For a given power series $\sum_{n=0}^{\infty} c_n (x-a)^n$, there are only three possibilities:

  1. The series converges only when $x = a$.
  2. The series converges for all $x$.
  3. There is a positive number $R$ such that the series converges if $|x - a| < R$ and diverges if $|x - a| > R$.

The number $R$ is called the radius of convergence. If the series converges only when $x = a$, then $R = 0$. If the series converges for all $x$, then $R = \infty$. The interval of convergence is the interval that consists of all values of $x$ for which the series converges.

Section 11.9: Representations of Functions as Power Series

We can often express functions as power series. For instance, recall the geometric series:

$$ \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + ... $$

This representation is valid for $|x| < 1$. We can use this to find power series representations of other functions.

Example: Express $f(x) = \frac{1}{1 + x^2}$ as a power series. Replacing $x$ with $-x^2$ in the geometric series, we get:

$$ \frac{1}{1 + x^2} = \frac{1}{1 - (-x^2)} = \sum_{n=0}^{\infty} (-x^2)^n = \sum_{n=0}^{\infty} (-1)^n x^{2n} = 1 - x^2 + x^4 - x^6 + ... $$

This series converges when $|-x^2| < 1$, which means $x^2 < 1$, or $|x| < 1$. The interval of convergence is $(-1, 1)$.

Differentiation and Integration of Power Series:

If we have a power series representation for a function, we can differentiate and integrate the series term-by-term within its interval of convergence. This provides a powerful tool for finding representations of related functions.

Keep practicing, and you'll master these concepts in no time!