Integration of Power Series PDF

### DATE ### Integration of Power Series Let $f(x) = \sum_{n=0}^{\infty} C_n(x-a)^n$ have radius of convergence $R>0$. Then an antiderivative for $f$ is given by: $\int f(x) dx = [\sum_{n=0}^{\infty} \frac{C_n(x-a)^{n+1}}{n+1}] + C$ $x \in (a-R, a+R)$ ### Taylor and Maclaurin Series Let $f(x) =...

### DATE ### Integration of Power Series Let $f(x) = \sum_{n=0}^{\infty} C_n(x-a)^n$ have radius of convergence $R>0$. Then an antiderivative for $f$ is given by: $\int f(x) dx = [\sum_{n=0}^{\infty} \frac{C_n(x-a)^{n+1}}{n+1}] + C$ $x \in (a-R, a+R)$ ### Taylor and Maclaurin Series Let $f(x) = \sum_{n=0}^{\infty} a_nx^n$ w/ radius of conv $R>0$. $f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4 + ...$ $f'(x) = a_1 + 2a_2x + 3a_3x^2 + 4a_4x^3 + ...$ $f''(x) = 2a_2 + 6a_3x + 12a_4x^2 + ...$ $f'''(x) = 6a_3 + 24a_4 x+ ...$ $\rightarrow$ $f(0) = a_0$ $ f'(0) = a_1$ $f''(0) = a_2$/ 2 $a_2$ = $f''(0)/2$ $f'''(0) = a_3$/ $6a_3$ : $a_3 = f'''(0)/6$ $a_n = \frac{f^n(0)}{n!}$ **Def'n:** The Maclaurin Series for $f(x)$ is a power series $\sum_{n=0}^{\infty} \frac{f^n(0)}{n!} x^n = f(0) + f'(0)x + \frac{f''(0)}{2}x^2 + \frac{f'''(0)}{6}x^3 + \frac{f''''(0)}{24}x^4 + ...$ *Sterling*