Appendix D Some Taylor Series
Reference.
OpenStax Calculus Volume 2, Section 6.4 1| Function | Maclaurin Series | Interval and Radius of Convergence |
|---|---|---|
| \(f(x) = \ds\frac{1}{1-x}\) | \(\ds\sum_{n=0}^\infty x^n\) | \(-1 < x < 1\text{,}\) \(R=1\) |
| \(f(x) = e^x\) | \(\ds\sum_{n=0}^\infty \frac{x^n}{n!}\) | \(-\infty < x < \infty\text{,}\) \(R=\infty\) |
| \(f(x) = \ln(1+x)\) | \(\ds\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}x^n\) | \(-1 < x \le 1\text{,}\) \(R=1\) |
| \(f(x) = \sin(x)\) | \(\ds\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1}\) | \(-\infty < x < \infty\text{,}\) \(R=\infty\) |
| \(f(x) = \cos(x)\) | \(\ds\sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}x^{2n}\) | \(-\infty < x < \infty\text{,}\) \(R=\infty\) |
| \(f(x) = \arctan(x)\) | \(\ds\sum_{n=0}^\infty \frac{(-1)^n}{2n+1}x^{2n+1}\) | \(-1 \le x \le 1\text{,}\) \(R=1\) |
| \(f(x) = (1+x)^r\) | \(\ds\sum_{n=1}^\infty \left(\begin{array}{c} r \\ n \end{array}\right) x^n\) | \(-1 < x < 1\text{,}\) \(R=1\) (see note) |
| where \(\ds \left(\begin{array}{c} r \\ n \end{array}\right) = \frac{r(r-1) \cdots (r-n+1)}{n!}\) | ||
Note: the Binomial Series for \((1+x)^r\) also converges for all \(x\) if \(r\) is a natural number (where the series is just a polynomial), and also can converge at the endpoints \(x = \pm 1\) for some other values of \(r\text{.}\)
https://openstax.org/books/calculus-volume-2/pages/6-4-working-with-taylor-series