Class 26
\(\displaystyle \frac{1}{1-x} = \sum_{n=0}^\infty x^n = 1 + x + x^2 + x^3 + \cdots\) for \(-1 < x < 1\).
\(\displaystyle \frac{1}{1+x^2} = \sum_{n=0}^\infty (-1)^n x^{2n} = 1 - x^2 + x^4 - x^6 + \cdots\) for \(-1 < x < 1\).
\(\displaystyle \frac{1}{(1-x)^2} = \sum_{n=0}^\infty (n+1)x^n = 1 + 2x + 3x^2 + 4x^3 + \cdots\) for \(-1 < x < 1\).
\(\displaystyle \ln(1+x) = \sum_{n=1}^\infty (-1)^{n+1} \frac{x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots\) for \(-1 < x \le 1\).
\(\displaystyle \arctan(x) = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots\) for \(-1 \le x \le 1\).
\(\displaystyle \exp(x) = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \cdots\) for any \(x\).
\(\displaystyle \sin(x) = \sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{6} + \frac{x^5}{120} - \cdots\) for any \(x\).
\(\displaystyle \cos(x) = \sum_{n=0}^\infty (-1)^n\frac{x^{2n}}{(2n)!} = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots\) for any \(x\).
\[\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n\]
Let \(f\) has \(n+1\) derivatives on an open interval \(I\) containing \(a\). Then
\[f(x) = \sum_{i=0}^n \frac{f^{(i)}(a)}{i!}(x-a)^i + R_n(x)\]
and for every \(x \in I\) there exists a number \(c\) between \(a\) and \(x\) such that
\[R_n(x) = \frac{f^{(n+1)}({\color{red} c})}{(n+1)!}(x-a)^{n+1}\]
\(\displaystyle\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n = f(x)\) if and only if \(\displaystyle\lim_{n\to\infty}R_n(x) = 0\).
Expand \((1+x)^n\)
\(f(x) = (1 + x)^r\)
\(\sqrt{1+x}\)
Find a function \(f\) such that \(f'(x) = f(x)\).
Find \(\displaystyle \int \exp\left(-x^2\right)\;dx\)
Find \(\displaystyle\lim_{x\to 0}\frac{\sin x}{x}\)
Find \(\displaystyle\lim_{x\to 0}\frac{\cos x - 1}{x^2}\)
\(\exp (ix) = {}\)