Calculus Notes

Section 9.8 Maclaurin and Taylor Series; Power Series

If take a Taylor Polynomial to infinity we get a Taylor Series and can be used to approximate other functions. Some useful Taylor Series are:

• \(\displaystyle \sin(x) = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dfrac{x^7}{7!} + \dots\)

• \(\displaystyle \cos(x) = 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \dfrac{x^6}{6!} + \dots\)

• \(\displaystyle e^x = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \dfrac{x^4}{4!} + \dots\)

• \(\displaystyle \dfrac{1}{1 - x} = 1 + x + x^2 + x^3 + x^4 + \dots \quad \text{if } \mid x \mid \lt 1 \)

  Note 9.8.1.

  If we look at a geometric series where \(a = 1\) and \(x = x\text{,}\) we get:

  \begin{equation*} 1 + x + x^2 + x^3 + x^4 + \dots \end{equation*}

  Plugging this information into the geometric sum equation we get \(\dfrac{1}{1 - x}\) and we can see where we get this last series.

For the values of \(x\) for which the series converges to \(f(x)\) we have:

• Taylor sieres for \(f(x)\) about \(x = a\text{:}\)

  \begin{equation*} f(x) = f(a) + f'(a)(x - a) + \dfrac{f''(a)}{2!}(x = a)^2 + \dots + \dfrac{f^{(n)}(a)}{n!}(x - a)^n + \dots \end{equation*}

  This series is also called a Maclaurin series if \(a = 0\text{.}\)

• Binomial series:

  \begin{equation*} f(x) = (1 + x)^p = 1 + px + \dfrac{p(p - 1)x^2}{2!} + \dfrac{p(p - 1)(p - 2)x^3}{3!} + \dots \quad \text{if } \mid x \mid \lt 1 \text{, for any constant } p \end{equation*}

A power series about \(x = a\) is a sum of a constant times powers of \((x - a)\text{:}\)

where \(C_0, C_1, C_2, \text{ and } C_3\) are all constants.