What are some good or neat examples of computing a function's Taylor series?

Question

Starting from \begin{equation*} \frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots, \end{equation*} which holds when $|x| < 1$, we can conclude that \begin{equation*} \frac{1}{1 + x^2} = 1 - x^2 + x^4 - x^6 + \cdots \end{equation*} when $|x| < 1$. Now taking antiderivatives of both sides, we discover that \begin{equation*} \tan^{-1}(x) = C + x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \end{equation*} when $|x| < 1$. By plugging in $x = 0$, we see that $C = 0$, and we have found the Taylor series (centered at $0$) for $\tan^{-1}$.

This is a pretty neat calculation. Computing the derivatives of $\tan^{-1}$ at $0$ directly would have been laborious. What are some other neat or good examples of calculations like this (where a Taylor series is computed in an efficient way)?

A couple other examples would be computing the Maclaurin series for $x \sin(x)$ by just multiplying the Maclaurin series for $\sin(x)$ by $x$, or computing the Maclaurin series for $\sin(x) \cos(x)$ by multiplying the series for $\sin(x)$ by the series for $\cos(x)$. (You could also use the identity $\sin(x) \cos(x) = \frac12 \sin(2x)$.)

Answer 1

One of my favorites also provides a useful way to do an integral.

From the usual Taylor Series of $$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$ we can plug in $x^2$ to obtain $$e^{x^2} = \sum_{n=0}^{\infty} \frac{x^{2n}}{n!}.$$ As a note, this convergence is uniform on all compact subsets of $\mathbb{R}$ as the radius of convergence $R$ is $\infty$ (using the usual analytic notation).

Now, I know in lower level calculus classes that $\int e^{x^2}dx$ is mentioned to not have a closed form solution, that is, it is not a derivative of some composition of 'standard' functions. The issue lies in that many students do have to integrate on intervals to obtain values when they have to study normal distributions. So instead of yielding to pre-computed tables, it can be analytically computed, using uniform convergence where necessary or viewing as the generic antiderivative we teach calculus students, as such:

\begin{align*} \int e^{x^2} dx &= \int \sum_{n=0}^{\infty} \frac{x^{2n}}{n!} dx \\ &= \sum_{n=0}^{\infty} \int \frac{x^{2n}}{n!} dx \\ &= \sum_{n=0}^{\infty} \frac{x^{2n+1}}{n! (2n+1)} \end{align*}

Furthermore, we can do computations: \begin{align*} \int_a ^b e^{x^2} dx &\approx \sum_{n=0}^{N} \frac{b^{2n+1} - a^{2n+1}}{n! (2n+1)} \end{align*} with error $|\dfrac{d^{N+1} e^{x^2}}{dx^{N+1}} \dfrac{b^{N+1}-a^{N+1}}{(N+1)!}|$ for a $x \in [a,b]$.

Answer 2

$f(x) = \begin{cases}\exp(-\frac1{x^2}) & x \neq 0 \cr 0 & x = 0 \end{cases}$

If only to disabuse the students of the idea that smooth implies analytic.

The Fabius function or $F(x) := \sum_{n=0}^\infty \frac{\exp(i2^nx)}{n!}$ (or its real part) would be better examples since they're nowhere analytic, but that's a bit harder to verify. I have an answer over on MSE attempting to show non-analyticity of $F$, but I might have gotten the application of the Cauchy–Hadamard formula wrong.