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Taylor Series for \(f(x)\) at \(x_0\) \[ f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n = f(x_0) + f'(x_0)(x-x_0) + \frac{f''(x_0)}{2!}(x-x_0)^2 + \cdots \] when \(x_0=0\), the taylor series is also named as MacLaurin series: \[ f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots \]
If the function \(f(x)\) has infinite level of derivatives, it can be expanded into Taylor series if and only if \(\displaystyle \lim_{n\rightarrow\infty} R_n(x)=0\)
Proof. Let \(f(x) = S_{n+1}(x) + R_n(x)\) \[ \lim_{n\rightarrow\infty} R_n(x) = \lim_{n\rightarrow\infty}[f(x)-S_{n+1}(x)] = 0 \]
If \(f(x)\) can be expanded into power series, the expansion is unique, and it’s the same as its MacLaurin series.
Proof. Assume the \(f(x)\) can be expanded into power series \[ f(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n + \cdots \] Then we have \[ \begin{align} & f'(x) = a_1 + 2a_2x + \cdots + na_nx^{n-1} + \cdots \\ & f''(x) = 2!a_2 + \cdots + n(n-1)a_nx^{n-2} + \cdots \\ & f^{(n)}(x) = n!a_n \end{align} \]
Two methods to expand a function into power series:
Example. Expand \(\e^x\) into power series. \[ \e^x = 1 + x + \frac{1}{2!}x^2 + \frac{1}{3!}x^3 + \cdots + \frac{1}{n!}x^n + \cdots \] To find the radius of convergence, we compute the limit \[ \displaystyle \rho = \lim_{n\rightarrow\infty} \frac{n!}{(n+1)!} = \lim_{n\rightarrow\infty} \frac{1}{n+1} = 0 \] Hence the radius of convergence \(R = \infty\).
For any number \(x\), the remainder \[ \lim_{n\rightarrow\infty} |R_n(x)| = \lim_{n\rightarrow\infty}\left| \frac{\e^\xi}{(x+1)!}x^{n+1} \right| < \lim_{n\rightarrow\infty}\e^{|x|} \frac{|x|^{n+1}}{(n+1)!} = 0 \]
Hence \(\displaystyle \e^x = 1 + x + \frac{1}{2!}x^2 + \frac{1}{3!}x^3 + \cdots + \frac{1}{n!}x^n + \cdots~~(x\in\mathbb{R})\)
Example. Expand \(\sin x\) into power series. \[ f^{(n)}(0) = \left\{\begin{array}{ll}\begin{align} &0, && n = 2k \\ &(-1)^k, && n = 2k+1 \end{align}\end{array}\right. \]
\[ \sin x = \sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^3}{5!} - \frac{x^7}{7!} + \cdots \] The radius of convergence \(R = \infty\). For any \(x\), the remainder \[ \lim_{n\rightarrow\infty} |R_n(x)| = \lim_{n\rightarrow\infty} \left| \frac{\sin\left(\xi + (n+1)\dfrac{\pi}{2}\right)}{(n+1)!} \right| < \lim_{n\rightarrow\infty} \frac{|x|^{n+1}}{(n+1)!} = 0 \]
\[ (1+x)^m = 1 + mx + \frac{m(m-1)}{2!}x^2 + \cdots + \frac{m(m-1)(m-n+1)}{n!}x^n + \cdots (-1<x<1) \] \(m = 1/2\), \[ \sqrt{1+x} = 1 + \frac{1}{2}x + \frac{1}{2\times4}x^2 + \frac{1\times3}{2\times4\times6}x^3 + \cdots \]
\[ \frac{1}{\sqrt{1+x}} = 1 + \frac{1}{2}x + \frac{1\times3}{2\times4}x^2 + \cdots \]
\[ \frac{1}{1+x} = 1 -x +x^2 -x^3 +\cdots+(-1)^nx^n +\cdots (-1<x<1) \]
\[ \frac{1}{1-x} = 1 +x +x^2 +x^3 + \cdots \]
Applying the known function power series expansion:
Example. Expand \(\dfrac{1}{1+x^2}\) \[ \frac{1}{1+x^2} = \]
Example. Expand \(\ln(1+x)\) into the power series of \(x\)
\(\displaystyle \ln(1+x) = \sum_{n=1}^{\infty}(-1)^n \int_0^x x^n = \sum_{n=1}^{\infty} \frac{(-1)^n}{n+1}x^{n+1}\)
\[ \ln 2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots \]
Example. Expand \(\sin x\) into power series of \(\left(x-\dfrac{\pi}{4}\right)\).