Taylor Series
Taylor Polynomial at $x=a$
- Constant: $f(x) \approx f(a)$
- Linear: $f(x) \approx f(a) + fâ(a)(x-a)$
- Quadratic $f(x) \approx f(a) + fâ(a)(x-a) + \frac{1}{2}fââ(a)(x-a)^2$
We write $f^{(n)}(x)$ to mean the $n$th derivative of $f(x)$, and $T_n(x)$ to mean the $n$th degree Taylor polynomial of $f(x)$.
\[\begin{aligned} T_n(x) &= f(a) + f'(a)(x-a) + \frac{1}{2}f''(a)(x-a)^2 + \cdots + \frac{1}{n!}f^{(n)}(a)(x-a)^n \\ &= \sum_{i=0}^n \frac{f^{(i)}(a)}{i!} (x-a)^i \end{aligned}\]If $a=0$, we call the function a Maclaurin polynomial.
\[\begin{aligned} T_n(x) &= f(0) + f'(0)x + \frac{1}{2}f''(0)x^2 + \cdots + \frac{1}{n!}f^{(n)}(0)x^n \\ &= \sum_{i=0}^n \frac{f^{(i)}(0)}{i!} x^i \end{aligned}\]Tayloar and Maclaurin series are expansions of infinite degrees.
- Taylor series: $f(x) = \sum_{i=0}^{\infty} \frac{f^{(i)}(a)}{i!} (x-a)^i$
- Maclaurin series: $f(x) = \sum_{i=0}^\infty \frac{f^{(i)}(0)}{i!} x^i$
Note that the derivatives are taken at $x=a$.
Important Taylor Series Expansions
| Function and Series | First Few Terms | Interval of Convergence |
|---|---|---|
| $e^x=\sum_{n=0}^\infty \frac{x^n}{n!}$ | $1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$ | $(-\infty, \infty)$ |
| $\frac{1}{1-x}=\sum_{n=0}^\infty x^n$ | $1+x+x^2+x^3+\cdots$ | $(-1,1)$ |
Algebra of Power Series
Let $f(x)=\sum_{n=0}^\infty a_nx^n$ and $g(x)=\sum_{n=0}^\infty b_nx^n$ converge absolutely for $\vert x\vert<R$, and let $h(x)$ be continuous.
- $f(x)\pm g(x) = \sum_{n=0}^\infty (a_n\pm b_n)x^n\quad$ for $\vert x\vert<R$.
- $f(x)\, g(x) = (\sum_{n=0}^\infty a_nx^n)(\sum_{n=0}^\infty b_nx^n) = \sum_{n=0}^\infty (a_0b_n + a_1b_{n-1} + \ldots + a_nb_0)x^n\quad$ for $\vert x\vert<R$.
- $f(h(x)) = \sum_{n=0}^\infty a_n (h(x))^n \quad$ for $\vert h(x)\vert <R$. We simply substitute $h(x)$ for $x$ in the series.