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Find the Maclaurin series of \( e^{x^{2}} \)

\( e^{x}=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}=1+\frac{x}{1 !}+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\cdots \)

Replace \(x\) by \( x^{2} \Rightarrow \)

\( e^{x^{2}}=1+\frac{x^{2}}{1!}+\frac{\left(x^{2}\right)^{2}}{2 !}+\frac{\left(x^{2}\right)^{3}}{3 !}+\cdots \)

\( e^{x^{2}}=\sum_{n=0}^{\infty} \frac{x^{2 n}}{n !} \)

Find a power series representation for \( f(x)=\frac{1}{l-x} \)

use long diviston: \( \frac{1}{1-x}=1+x+x^{2}+x^{3} \)

 

 

 

 

 

 

 

\( f(x)=\frac{1}{1-x}=1+x +x^{2}+x^{3} \cdots=\sum_{n=0}^{\infty} x^{n}\quad      (G \cdot s) \)

It is convergent only if \( |r|=|x|<1 \quad ∴ x \in(-1,1) \)

\(∴ \sum_{n=0}^{\infty} x^{n}=\frac{1}{1-x} \) , Interval of Convergence \( (1 . c)=(-1,1) \) 

                                Radius of Convergence \( =1 \)

Solution 2: using Maclaurin Series:

Maclaurin Series\( \frac{1}{1-x}=f(0)+\frac{f'{(0)}}{1 !}+\frac{f''{(0)}}{{2!}} x^{2} \)

                                    \( =1+\frac{x}{1 !}+\frac{2 x^{2}}{2 !}+\frac{(3)(2) x^{3}}{3!}+\cdots \)

                                    \( =1+x+x^{2}+x^{3}\ldots \)

\( \frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n} \) using Ratio test \( , u_{n}=x^{n} \)

\( u_{n}+1=x^{n+1} \ ,  \lim _{n \rightarrow \infty} |\frac{u_ n+1}{u_n}|= \lim _{n \rightarrow \infty} |\frac{x^{n+1}}{x^{n}}|= \lim _{n\rightarrow \infty}|x|=|x|<1 \)

\( \sum_{n=0}^{\infty} x^{n}= \frac{1}{1-x} \) is \((A . c)\) at \( {x} \in(-1, 1) \) ,  Raduis \(=1\)

Find the Taylor series for \( e^{x} \) at \(x=2\)

\( f^{n}{(2)}=e^{2} \)

\(c=2\)

Taylor Series definition:

\( e^{x}=\sum_{n=0}^{\infty} \frac{f^{n}{(2)}}{n !}(x-2)^{n}=\sum_{n=0}^{\infty} \frac{e^{x}}{n !}(x-2)^{n} \)

\( \lim _{n \rightarrow \infty}\left|\frac{u_{n}+1}{u_{n}}\right|= \lim _{n\rightarrow \infty}\left|\frac{e^{x}(x-2)^{n+1}}{(n+1) !} \cdot \frac{n !}{e^{x}(x-2)^{n}}\right| \)

                        \( =\lim _{n \rightarrow \infty} | \frac{(x-2) \cdot n !}{(n+1)(n)!}| =|x-2| \lim_ {n\rightarrow\infty} \left|\frac{1}{n !}\right| \)

                        \( =0<1 \)

\( A \cdot C \) for all \( x \in R \)

\( I \cdot C=(-\infty, \infty) \)

\( R \cdot C=\infty \)

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