a) The radius of convergence is calculated as
R=1.
b) Due to the fact that it converges in every direction, the radius of convergence is either infinity or zero.
<h3>What is the associated radius of convergence.?</h3>
(a)
Take into consideration the function f with respect to the number a,
![f(x)=\frac{1}{x}, \quad a=1](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7B1%7D%7Bx%7D%2C%20%5Cquad%20a%3D1)
In case you forgot, the Taylor series for the function $f$ at the number a looks like this:
![\begin{aligned}f(x) &=\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n !}(x-a)^{n} \\&=f(a)+\frac{f^{\prime}(a)}{1 !}(x-a)+\frac{f^{*}(a)}{2 !}(x-a)^{2}+\ldots\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7Df%28x%29%20%26%3D%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%20%5Cfrac%7Bf%5E%7B%28n%29%7D%28a%29%7D%7Bn%20%21%7D%28x-a%29%5E%7Bn%7D%20%5C%5C%26%3Df%28a%29%2B%5Cfrac%7Bf%5E%7B%5Cprime%7D%28a%29%7D%7B1%20%21%7D%28x-a%29%2B%5Cfrac%7Bf%5E%7B%2A%7D%28a%29%7D%7B2%20%21%7D%28x-a%29%5E%7B2%7D%2B%5Cldots%5Cend%7Baligned%7D)
Determine the function f as well as any derivatives of the function $f by setting a=1 and working backward from there.
![\begin{aligned}f(x) &=\frac{1}{x} & f(1)=\frac{1}{1}=1 \\\\f^{\prime}(x) &=-\frac{1}{x^{2}} & f^{\prime}(1)=-\frac{1}{(1)^{2}}=-1 \\\\f^{\prime \prime}(x) &=\frac{2}{x^{3}} & f^{\prime \prime}(1)=\frac{2}{(1)^{3}}=2 \\\\f^{\prime \prime}(x) &=-\frac{2 \cdot 3}{x^{4}} & f^{\prime \prime}(1)=-\frac{2 \cdot 3}{(1)^{4}}=-2 \cdot 3 \\\\f^{(*)}(x) &=\frac{2 \cdot 3 \cdot 4}{x^{5}} & f^{(n)}(1)=\frac{2 \cdot 3 \cdot 4}{(1)^{5}}=2 \cdot 3 \cdot 4\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7Df%28x%29%20%26%3D%5Cfrac%7B1%7D%7Bx%7D%20%26%20f%281%29%3D%5Cfrac%7B1%7D%7B1%7D%3D1%20%5C%5C%5C%5Cf%5E%7B%5Cprime%7D%28x%29%20%26%3D-%5Cfrac%7B1%7D%7Bx%5E%7B2%7D%7D%20%26%20%20f%5E%7B%5Cprime%7D%281%29%3D-%5Cfrac%7B1%7D%7B%281%29%5E%7B2%7D%7D%3D-1%20%5C%5C%5C%5Cf%5E%7B%5Cprime%20%5Cprime%7D%28x%29%20%26%3D%5Cfrac%7B2%7D%7Bx%5E%7B3%7D%7D%20%26%20%20f%5E%7B%5Cprime%20%5Cprime%7D%281%29%3D%5Cfrac%7B2%7D%7B%281%29%5E%7B3%7D%7D%3D2%20%5C%5C%5C%5Cf%5E%7B%5Cprime%20%5Cprime%7D%28x%29%20%26%3D-%5Cfrac%7B2%20%5Ccdot%203%7D%7Bx%5E%7B4%7D%7D%20%26%20f%5E%7B%5Cprime%20%5Cprime%7D%281%29%3D-%5Cfrac%7B2%20%5Ccdot%203%7D%7B%281%29%5E%7B4%7D%7D%3D-2%20%5Ccdot%203%20%5C%5C%5C%5Cf%5E%7B%28%2A%29%7D%28x%29%20%26%3D%5Cfrac%7B2%20%5Ccdot%203%20%5Ccdot%204%7D%7Bx%5E%7B5%7D%7D%20%26%20f%5E%7B%28n%29%7D%281%29%3D%5Cfrac%7B2%20%5Ccdot%203%20%5Ccdot%204%7D%7B%281%29%5E%7B5%7D%7D%3D2%20%5Ccdot%203%20%5Ccdot%204%5Cend%7Baligned%7D)
At the point when a = 1, the Taylor series for the function f looks like this:
![f(x) &=f(a)+\frac{f^{\prime}(a)}{1 !}(x-a)+\frac{f^{\prime \prime}(a)}{2 !}(x-a)^{2}+\frac{f^{\prime \prime \prime}(a)}{3 !}(x-a)^{3}+\cdots \\\\&=f(1)+\frac{f^{\prime}(1)}{1 !}(x-1)+\frac{f^{\prime}(1)}{2 !}(x-1)^{2}+\frac{f^{\prime \prime}(1)}{3 !}(x-1)^{3}+\cdots \\](https://tex.z-dn.net/?f=f%28x%29%20%26%3Df%28a%29%2B%5Cfrac%7Bf%5E%7B%5Cprime%7D%28a%29%7D%7B1%20%21%7D%28x-a%29%2B%5Cfrac%7Bf%5E%7B%5Cprime%20%5Cprime%7D%28a%29%7D%7B2%20%21%7D%28x-a%29%5E%7B2%7D%2B%5Cfrac%7Bf%5E%7B%5Cprime%20%5Cprime%20%5Cprime%7D%28a%29%7D%7B3%20%21%7D%28x-a%29%5E%7B3%7D%2B%5Ccdots%20%5C%5C%5C%5C%26%3Df%281%29%2B%5Cfrac%7Bf%5E%7B%5Cprime%7D%281%29%7D%7B1%20%21%7D%28x-1%29%2B%5Cfrac%7Bf%5E%7B%5Cprime%7D%281%29%7D%7B2%20%21%7D%28x-1%29%5E%7B2%7D%2B%5Cfrac%7Bf%5E%7B%5Cprime%20%5Cprime%7D%281%29%7D%7B3%20%21%7D%28x-1%29%5E%7B3%7D%2B%5Ccdots%20%5C%5C)
![&=1+\frac{-1}{1 !}(x-1)+\frac{2}{2 !}(x-1)^{2}+\frac{-2 \cdot 3}{3 !}(x-1)^{3}+\frac{2 \cdot 3 \cdot 4}{4 !}(x-1)^{4}+\cdots \\\\&=1-(x-1)+(x-1)^{2}-(x-1)^{3}+(x-1)^{4}+\cdots \\\\&=\sum_{1=0}^{\infty}(-1)^{n}(x-1)^{n}](https://tex.z-dn.net/?f=%26%3D1%2B%5Cfrac%7B-1%7D%7B1%20%21%7D%28x-1%29%2B%5Cfrac%7B2%7D%7B2%20%21%7D%28x-1%29%5E%7B2%7D%2B%5Cfrac%7B-2%20%5Ccdot%203%7D%7B3%20%21%7D%28x-1%29%5E%7B3%7D%2B%5Cfrac%7B2%20%5Ccdot%203%20%5Ccdot%204%7D%7B4%20%21%7D%28x-1%29%5E%7B4%7D%2B%5Ccdots%20%5C%5C%5C%5C%26%3D1-%28x-1%29%2B%28x-1%29%5E%7B2%7D-%28x-1%29%5E%7B3%7D%2B%28x-1%29%5E%7B4%7D%2B%5Ccdots%20%5C%5C%5C%5C%26%3D%5Csum_%7B1%3D0%7D%5E%7B%5Cinfty%7D%28-1%29%5E%7Bn%7D%28x-1%29%5E%7Bn%7D)
In conclusion,
![&=\sum_{1=0}^{\infty}(-1)^{n}(x-1)^{n}](https://tex.z-dn.net/?f=%26%3D%5Csum_%7B1%3D0%7D%5E%7B%5Cinfty%7D%28-1%29%5E%7Bn%7D%28x-1%29%5E%7Bn%7D)
Find the radius of convergence by using the Ratio Test in the following manner:
![\begin{aligned}L &=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right| \\&=\lim _{n \rightarrow \infty} \frac{(-1)^{n+1}(x-1)^{n+1}}{(-1)^{n}(x-1)^{n}} \mid \\&=\lim _{n \rightarrow \infty}|x-1| \\&=|x-1|\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7DL%20%26%3D%5Clim%20_%7Bn%20%5Crightarrow%20%5Cinfty%7D%5Cleft%7C%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_%7Bn%7D%7D%5Cright%7C%20%5C%5C%26%3D%5Clim%20_%7Bn%20%5Crightarrow%20%5Cinfty%7D%20%5Cfrac%7B%28-1%29%5E%7Bn%2B1%7D%28x-1%29%5E%7Bn%2B1%7D%7D%7B%28-1%29%5E%7Bn%7D%28x-1%29%5E%7Bn%7D%7D%20%5Cmid%20%5C%5C%26%3D%5Clim%20_%7Bn%20%5Crightarrow%20%5Cinfty%7D%7Cx-1%7C%20%5C%5C%26%3D%7Cx-1%7C%5Cend%7Baligned%7D)
The convergence of the series when L<1, that is, |x-1|<1.
The radius of convergence is calculated as
R=1.
For B
Take into consideration the function f with respect to the number a,
![a_{n}=(-1)^{n}(x-1)^{n}](https://tex.z-dn.net/?f=a_%7Bn%7D%3D%28-1%29%5E%7Bn%7D%28x-1%29%5E%7Bn%7D)
The Taylor series for f(x)=e^{x} at a=0 is,
![e^{2}=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}+\ldots](https://tex.z-dn.net/?f=e%5E%7B2%7D%3D1%2Bx%2B%5Cfrac%7Bx%5E%7B2%7D%7D%7B2%20%21%7D%2B%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%20%21%7D%2B%5Cfrac%7Bx%5E%7B4%7D%7D%7B4%20%21%7D%2B%5Cldots)
![f(x) &=\left(x^{2}+2 x\right) e^{x} \\&=\left(x^{2}+2 x\right)\left(1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}+\ldots\right)+2 x\left(1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}+\ldots\right) \\&=x^{2}\left(1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}+\ldots\right)+\left(\frac{x^{4}}{2 !}+\frac{x^{5}}{3 !}+\frac{x^{6}}{4 !}+\ldots\right)+\left(2 x+2 x^{2}+\frac{2 x^{3}}{2 !}+\frac{2 x^{4}}{3 !}+\frac{2 x^{5}}{4 !}+\ldots\right) \\](https://tex.z-dn.net/?f=f%28x%29%20%26%3D%5Cleft%28x%5E%7B2%7D%2B2%20x%5Cright%29%20e%5E%7Bx%7D%20%5C%5C%26%3D%5Cleft%28x%5E%7B2%7D%2B2%20x%5Cright%29%5Cleft%281%2Bx%2B%5Cfrac%7Bx%5E%7B2%7D%7D%7B2%20%21%7D%2B%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%20%21%7D%2B%5Cfrac%7Bx%5E%7B4%7D%7D%7B4%20%21%7D%2B%5Cldots%5Cright%29%2B2%20x%5Cleft%281%2Bx%2B%5Cfrac%7Bx%5E%7B2%7D%7D%7B2%20%21%7D%2B%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%20%21%7D%2B%5Cfrac%7Bx%5E%7B4%7D%7D%7B4%20%21%7D%2B%5Cldots%5Cright%29%20%5C%5C%26%3Dx%5E%7B2%7D%5Cleft%281%2Bx%2B%5Cfrac%7Bx%5E%7B2%7D%7D%7B2%20%21%7D%2B%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%20%21%7D%2B%5Cfrac%7Bx%5E%7B4%7D%7D%7B4%20%21%7D%2B%5Cldots%5Cright%29%2B%5Cleft%28%5Cfrac%7Bx%5E%7B4%7D%7D%7B2%20%21%7D%2B%5Cfrac%7Bx%5E%7B5%7D%7D%7B3%20%21%7D%2B%5Cfrac%7Bx%5E%7B6%7D%7D%7B4%20%21%7D%2B%5Cldots%5Cright%29%2B%5Cleft%282%20x%2B2%20x%5E%7B2%7D%2B%5Cfrac%7B2%20x%5E%7B3%7D%7D%7B2%20%21%7D%2B%5Cfrac%7B2%20x%5E%7B4%7D%7D%7B3%20%21%7D%2B%5Cfrac%7B2%20x%5E%7B5%7D%7D%7B4%20%21%7D%2B%5Cldots%5Cright%29%20%5C%5C)
![&=\left(x^{2}+x^{3}+\frac{x^{4}}{2 !}\right) \\&=2 x+3 x^{2}+\left(1+\frac{2}{2 !}\right) x^{3}+\left(\frac{1}{2 !}+\frac{2}{3 !}\right) x^{4}+\left(\frac{1}{4 !}\right) x^{5}+\ldots \\&=2 x+3 x^{2}+2 x^{3}+\frac{5}{6} x^{4}+\frac{1}{4} x^{5}+\ldots](https://tex.z-dn.net/?f=%26%3D%5Cleft%28x%5E%7B2%7D%2Bx%5E%7B3%7D%2B%5Cfrac%7Bx%5E%7B4%7D%7D%7B2%20%21%7D%5Cright%29%20%5C%5C%26%3D2%20x%2B3%20x%5E%7B2%7D%2B%5Cleft%281%2B%5Cfrac%7B2%7D%7B2%20%21%7D%5Cright%29%20x%5E%7B3%7D%2B%5Cleft%28%5Cfrac%7B1%7D%7B2%20%21%7D%2B%5Cfrac%7B2%7D%7B3%20%21%7D%5Cright%29%20x%5E%7B4%7D%2B%5Cleft%28%5Cfrac%7B1%7D%7B4%20%21%7D%5Cright%29%20x%5E%7B5%7D%2B%5Cldots%20%5C%5C%26%3D2%20x%2B3%20x%5E%7B2%7D%2B2%20x%5E%7B3%7D%2B%5Cfrac%7B5%7D%7B6%7D%20x%5E%7B4%7D%2B%5Cfrac%7B1%7D%7B4%7D%20x%5E%7B5%7D%2B%5Cldots)
Due to the fact that it converges in every direction, the radius of convergence is either infinity or zero.
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