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Reference Materials for Mathematics

Remkes Kooistra

Section 4.8 Series

Subsection 4.8.1 Geometric Series and \(\zeta\) Series

The geometric series converges if \(r\text{,}\) the common ratio, satisfies \(|r| \lt 1\text{.}\) We also know the value of the series.
\begin{equation*} \sum_{k=0}^\infty r^k = \frac{1}{1-r} \text{ if } |r| \lt 1 \end{equation*}
The \(\zeta\) series converges if \(p \gt 1\text{,}\) though we do not have an easy expression for its value.
\begin{equation*} \sum_{k=0}^\infty \frac{1}{k^p} \text{ converges if } p > 1 \end{equation*}

Subsection 4.8.2 Taylor Series

For real numbers \(\alpha, \ c \in \RR\text{,}\) assume that \(f(x)\) is infinitely differentiable on the domain \((\alpha - c, \alpha + c)\text{.}\) Then \(f\)has a Taylor series centered at \(\alpha\text{.}\)
\begin{equation*} f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(\alpha)}{n!} (x-\alpha)^n \end{equation*}
If all the \(a_n \neq 0\text{,}\) then the radius of convergence \(R\) is at least \(c\text{,}\) and is calculated by the following expression. If some of the coefficients are zero, then the ratio test is used to calculate the radius of convergence. If \(R=\infty\text{,}\) the Taylor series converges for all \(x \in \RR\text{.}\)
\begin{equation*} R = \frac{1}{\lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n} \right| } = \frac{1}{\lim_{n \rightarrow \infty} \sqrt[n]{|a_n|}} \text{ where } a_n = \frac{f^{(n)}(\alpha)}{n!} \end{equation*}

Subsection 4.8.3 Calculus of Taylor Series

Assume that \(f\) has a convergent Taylor series with radius of convergence \(R\text{,}\) possibly infinite. Then the integral and the derivative of the Taylor series can be calculated term-by-term.
\begin{align*} \int \sum_{n=0}^\infty a_n x^n dx \amp = \sum_{n=0}^\infty \int a_n x^n dx = \sum_{n=0}^\infty \frac{a_n x^{n+1}}{n+1} + C \\ \frac{d}{dx} \sum_{n=0}^\infty a_n x^n \amp = \sum_{n=0}^\infty \frac{d}{dx} a_n x^n = \sum_{n=1}^\infty a_n nx^{n-1} \end{align*}

Subsection 4.8.4 Common McLaurin Series

A MacLaurin series is a Taylor series centred at zero. Here are the McLaurins series for a number of commonly used functions, along with their radii of convergence.
\begin{align*} \amp \text{Function} \amp \amp \text{Taylor Series} \amp \amp \text{Radius of Convergence} \\ \amp e^x \amp \amp \displaystyle \sum_{n=0}^\infty \frac{x^n}{n!} \amp \amp \infty\\ \amp \sin x \amp \amp \displaystyle \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} \amp \amp \infty \\ \amp \cos x \amp \amp \displaystyle \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} \amp \amp \infty \\ \amp \arctan x \amp \amp \displaystyle \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{2n+1} \amp \amp \infty \\ \amp \frac{1}{1-x} \amp \amp \displaystyle \sum_{n=0}^\infty x^n \amp \amp 1 \\ \amp \ln (1-x) \amp \amp \displaystyle -\sum_{n=1}^\infty \frac{x^n}{n} \amp \amp 1 \\ \amp \ln (1+x) \amp \amp \displaystyle \sum_{n=1}^\infty (-1)^{n+1} \frac{x^n}{n} \amp \amp 1 \\ \amp (1+x)^r \amp \amp \displaystyle \sum_{n=0}^\infty \binom{r}{n} x^n \amp \amp \infty\\ \amp \sinh x \amp \amp \displaystyle \sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)!} \amp \amp \infty \\ \amp \cosh x \amp \amp \displaystyle \sum_{n=0}^\infty \frac{x^{2n}}{(2n)!} \amp \amp \infty \\ \amp \arctanh \ x \amp \amp \displaystyle \sum_{n=0}^\infty \frac{x^{2n+1}}{2n+1} \amp \amp \infty \\ \amp \text{Li}_k(x) \amp \amp \displaystyle \sum_{n=0}^\infty \frac{x^n}{n^k} \amp \amp 1 \\ \amp J_k(x) \amp \amp \displaystyle \sum_{n=0}^\infty \frac{(-1)^n x^{2n+k}}{n!(n+k)!2^{2n+k}} \amp \amp \infty \end{align*}
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