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• Covered today: Taylor series
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Find Maclaurin series representation of the function. On what interval each representation valid?

1. $e^{3x+1}$
2. $x^{2}\sin(x)$
3. $\ln\left(\frac{1-x}{1+x}\right)$
4. $\sqrt{x+1}$
1. Show that the function
   

$$f(x) = \left\{ \begin{array}{ll} e^{-\frac{1}{x^{2}}} & \mbox{ if $x\ne 0$} \\ 0 & \mbox{ if $x= 0$} \end{array} \right.$$

has derivative of all orders at every point of the real line, and $f^{(k)}(0)=0$ for every integer $k$. What is the interval of convergence of the Maclaurin series representation of the function? On what interval does the series converge to $f(x)$?

Find the Taylor series about the indicated center and determine the interval of convergence.

1. $f(x)=e^{x-1}$, $c=1$
2. $f(x)=e^{x}$, $c=2$
3. $f(x)=\ln(x)$, $c=e$

Use Taylor series to verify the given formulas:

1. $\sum_{k=0}^{\infty}\frac{2^{k}}{k!}=e^{2}$
2. $\sum_{k=0}^{\infty}\frac{(-1)^{k}\pi^{2k+1}}{(2k+1)!}=0$
3. $\sum_{k=0}^{\infty}\frac{(-1)^{k}}{2k+1}=\frac{\pi}{4}$
4. $\sum_{k=0}^{\infty}\frac{(-1)^{k+1}2^{k}}{k}=\ln(3)$