• Over the next few sections, we will be learning about power series, Taylor series, and Taylor polynomials.
• You might be asking "why are we learning about these things? And how does this relate to all of the things we have learned about in the past 5 sections?"
• Let's start with an example that will hopefully motivate our discussion over the next 3 sections.
• Take a few minutes to compute the following:
  
  \(\displaystyle \int e^{-x^2} \, dx\).

• You couldn't compute \(\int e^{-x^2} \, dx\).
• It turns you can't express the antiderivative of \(e^{-x^2}\) using elementary functions.
• However, it can be shown that
  
  \(\displaystyle e^{-x^2} = \sum_{n=0}^\infty \frac{(-x^2)^n}{n!} = \sum_{n=0}^\infty (-1)^n\frac{x^{2n}}{n!}\)
  \(\displaystyle \, \, \, \, \, \, = 1 - x^2 + \frac{x^4}{2} - \frac{x^6}{6}+\frac{x^8}{24} - \frac{x^{10}}{120}+\cdots\)
• The expression on the right is called a power series.
• It is also called the Taylor series for \(e^{-x^2}\) centered at 0.
• We will see later why these are equal.

• We have that
  
  \(\displaystyle e^{-x^2} = \sum_{n=0}^\infty \frac{(-x^2)^n}{n!} = \sum_{n=0}^\infty (-1)^n\frac{x^{2n}}{n!}\)
• Let's look at some graphs to see what is going on: Desmos e^(-x^2) Plot

• We have that
  
  \(\displaystyle e^{-x^2} = \sum_{n=0}^\infty \frac{(-x^2)^n}{n!} = \sum_{n=0}^\infty (-1)^n\frac{x^{2n}}{n!}\)
• Notice that
  
  \(\displaystyle \lim_{n\rightarrow\infty} \left|\frac{(-x^2)^{n+1}}{(n+1)!}\cdot\frac{n!}{(-x^2)^n}\right| =\lim_{n\rightarrow\infty} \left|\frac{-x^2}{n+1}\right| =0 \)
  
  no matter what \(x\) is.
• So the radius of convergence is \(\infty\).
• That means that the power series "works" everywhere.

• Now, if \(\displaystyle e^{-x^2} = 1 - x^2 + \frac{x^4}{2} - \frac{x^6}{6}+\frac{x^8}{24} - \frac{x^{10}}{120}+\cdots\)
• It stands to reason that
  
  \(\displaystyle \int e^{-x^2} \, dx = x - \frac{x^3}{3} + \frac{x^5}{10} - \frac{x^7}{42}+\frac{x^9}{216} - \frac{x^{11}}{1320}+\cdots\)
• Thus, we kind of have a way to express the antiderivative of \(e^{-x^2}\) using elementary functions. So what's the problem?
• Given a value of \(x\), how do you compute \(\displaystyle \sum_{n=0}^\infty (-1)^n\frac{x^{2n}}{n!}\)?

• How do we express a given function as a power series?
• If we express a function as a power series, for what values of \(x\) is it valid?
• How do we evaluate a power series at a given value? (It has infinitely many terms.)
• Can we estimate the value of a power series at a given value? If so, how do we know how accurate our estimate is?

Recall from calc 1: the derivative of a function at a single point \(f'(a)\) is a special case of the derivative function \(f'(x)\) obtained by evaluating at \(x=a\).

Now in calc 2: the sum of a series \(\displaystyle \sum_{n=0}^{\infty} a_n\) is a special case of the sum of a power series \(f(x) = \displaystyle \sum_{n=0}^{\infty} a_n x^n\) obtained by evaluating at \(x=1\).

Given a power series, does it converge to function we already know and love? For what \(x\) does the series converge?

Definition: A power series in \(x\) centered at \(0\) is a function of the form \[f(x) = \sum_{n=0}^{\infty} a_n x^n = a_0 + a_1 x + a_2 x^2 + \cdots.\] A power series in \(x\) centered at \(c\) is a function of the form \[f(x) = \sum_{n=0}^{\infty} a_n (x-c)^n = a_0 + a_1(x-c) + a_2(x-c)^2 + \cdots.\]

Let \(\displaystyle f(x) = \sum_{n=0}^{\infty} x^n\).

1. Is this a power series? If so, what is \(a_n\)?

2. For what values of \(x\) does \(f(x)\) converge?

3. For what values of \(x\) does \(f(x-4)\) converge?

4. For what values of \(x\) does \(\frac{1}{5}f(x/5)\) converge?

\(\displaystyle f(x) = \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}\) for \(-1 < x < 1\)

\(\displaystyle f(x-4) = \sum_{n=0}^{\infty} (x-4)^n = \frac{1}{1-(x-4)} = \frac{1}{5-x}\) for \(3 < x < 5\)

\(\displaystyle \frac{1}{5} f\left(\frac{x}{5}\right) = \frac{1}{5} \sum_{n=0}^{\infty} \left(\frac{x}{5}\right)^n = \frac{1}{5} \cdot \frac{1}{1-\frac{x}{5}} = \frac{1}{5-x}\) for \(-5 < x < 5\)

• See 10.6.4 Radius Of Convergence Theorem
• See 10.6.6 Radius and Interval Of Convergence
• See 10.6.7 Radius Of Convergence and Absolute Convergence

Find the interval of all \(x\) values for which the series converges. That is, find the center, determine the radius of convergence using the ratio test, and check for convergence at the points where the ratio test is inconclusive.

1. \(\displaystyle \sum_{n=1}^{\infty} \frac{(x+1)^n}{n \cdot 3^n}\). Remember to check endpoints!

2. \(\displaystyle \sum_{n=0}^\infty \frac{(-2)^n x^n}{(2n)!}\). What do you think this converges to? Plot a partial sum with many terms!

\(\displaystyle \cos(\sqrt{2x}) = \sum_{n=0}^\infty \frac{(-2x)^n}{(2n)!}\) for \(x \geq 0\).

On the interval of convergence \(I\) of a power series \(\displaystyle f(x) = \sum_{n=0}^{\infty} a_n (x-c)^n\),

1. If \(f\) is differentiable, you can compute its derivative term-by-term and the result will coverge on \(I\).

2. If \(f\) is continuous, you can compute its integral term-by-term and the result will converge on \(I\) (up to a constant \(+C\), of course).

We know \(\displaystyle f(x) = \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}\) for \(-1 < x < 1\).

1. Compute the derivative and integral of \(f(x)\), both as a closed-form expression and as a power series. For what values of \(x\) do the power series expressions for the derivative and integral converge to the closed-form expressions?

2. What is a power series representation for \(\displaystyle \frac{1}{1-x^2}\)? For \(\displaystyle \frac{1}{1+2x}\)? How does the interval of convergence change?