Appendix D Some Power Series
| \(f(x) = \ds\frac{1}{1-x}\) |
\(\ds\sum_{n=0}^\infty x^n\) |
\(-1 < x < 1\text{,}\) \(R=1\)
|
| \(f(x) = e^x\) |
\(\ds\sum_{n=0}^\infty \frac{x^n}{n!}\) |
\(-\infty < x < \infty\text{,}\) \(R=\infty\)
|
| \(f(x) = \ln(1+x)\) |
\(\ds\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}x^n\) |
\(-1 < x \le 1\text{,}\) \(R=1\)
|
| \(f(x) = \sin(x)\) |
\(\ds\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1}\) |
\(-\infty < x < \infty\text{,}\) \(R=\infty\)
|
| \(f(x) = \cos(x)\) |
\(\ds\sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}x^{2n}\) |
\(-\infty < x < \infty\text{,}\) \(R=\infty\)
|
| \(f(x) = \arctan(x)\) |
\(\ds\sum_{n=0}^\infty \frac{(-1)^n}{2n+1}x^{2n+1}\) |
\(-1 \le x \le 1\text{,}\) \(R=1\)
|
| \(f(x) = (1+x)^r\) |
\(\ds\sum_{n=1}^\infty \left(\begin{array}{c} r \\ n \end{array}\right) x^n\) |
\(-1 < x < 1\text{,}\) \(R=1\) (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 \(r\text{.}\)
And all of these come from
Theorem D.0.1. Taylor's Theorem.
For a Taylor polynomial \(T_n(x)\) for \(f\) with center \(a\text{,}\) and any number \(x\) in the domain of \(f\text{,}\) there is a number \(c\) between \(x\) and \(a\) such that
\begin{equation}
R_N(x) = \frac{f^{(N+1)}(c)}{(N+1)!} (x-a)^{N+1}\tag{D.0.1}
\end{equation}
and
Theorem D.0.2. Taylor's Inequality, and convergence to \(f(x)\text{,}\) sometimes.
If for some numbers \(M\) and \(d>0\text{,}\) \(|x-a| \leq d\) ensures that \(|f^{(n)}(x)| \leq M\text{,}\) for all \(n\text{,}\) then the remainder of the Taylor series for \(f\) with center \(a\) satisfies
\begin{equation}
|R_N(x)| \leq \frac{M}{(N+1)!} |x-a|^{N+1} \quad \text{ for } |x-a| \leq d.\tag{D.0.2}
\end{equation}
If this is true, then for \(|x-a| \leq d\text{,}\) \(|R_N(x)| \to 0\) as \(N \to \infty\text{,}\) so \(T_N(x) \to f(x)\text{,}\) so that the value of the Taylor Series is \(f(x)\text{:}\)
\begin{equation}
f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x-a)^n
= f(a) + f'(a) (x-a) + \frac{f''(a)}{2} (x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!} (x-a)^n \cdots\tag{D.0.3}
\end{equation}
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