Power Series, Taylor/Maclaurin and n-th derivative

Question

a) Find the convergence radius of the power series $$\sum_\limits{n=1}^∞\frac{(n+1)^{n^2}}{3^nn^{n^2}}x^n $$

b)Find all the $x\in \mathbb{R}$ for which the power series $\sum_\limits{n=1}^∞\frac{(x+1)^n}{\sqrt{4^nn}}$ converge.

c)Using the Maclaurin expansion of $e^x$, find the power series of $g(x)=e^{5x^2}$ with center $x_0=0$. Also for every $n\in \mathbb{Z}$ find a formula for $g^{(n)}(0)$.

My work so far:

a) I found $R=\frac{3}{e}$ with $R={(\limsup_{n\rightarrow∞}\sqrt[n]{|a_n|})}^{-1}$.

b) From the ratio test and $n\rightarrow ∞$ we get $|(x+1)\frac{1}{2}|$. So the power series converge for $|(x+1)\frac{1}{2}|<1$. Thus $-3<x<1$ is the interval of convergence of the power series.

c) The power series of $g(x)$ using the Maclaurin expansion of $e^x$ is ${\sum_\limits{n=0}^ ∞}\frac{5^n}{n!}x^{2n}$

Is my work correct so far? I don't know what to do for the n-th derivative of $g(x)$. Any help would be appreciated.

Answer 1

The Maclaurin series for $f$ is: $$f(x)=\sum\limits_{n \in \mathbb{N}} \left.\frac{\mathrm{d}^nf(x)}{\mathrm{d}x^n}\right|_{x=0}\frac{x^n}{n!}$$ I think this will be enough for you to finish the excercise c.

So, the first few terms of the Maclaurin series are: $$g^{(0)}(0)\frac{x^0}{0!}+g^{(1)}(0)\frac{x^1}{1!}+g^{(2)}(0)\frac{x^2}{2!}+\dots$$ $$g^{(0)}(0)+g^{(1)}(0)x+g^{(2)}(0)\frac{x^2}{2}+\dots$$ While your function is: $$5^0\frac{x^{2*0}}{0!}+5^1\frac{x^{2*1}}{1!}+5^2\frac{x^{2*2}}{2!}+\dots$$ $$1+5x^2+\frac{25}{2}x^4+\dots$$

Answer 2

Finding the n-th derivative of $g$ is easy, since you have already found the Maclaurin expansion:

• Since $g$ is an even function, $g^{(n)}(0)=0$, if $n$ is odd.
• From the Maclaurin expansion, if $n$ is even:
  • $ \begin{align*} g(x) =\sum_{n=0}^{\infty}\frac{g^{(n)}(0)}{n!}x^n =\sum_{v=0}^\infty \frac{5^v}{v!}x^{2v}\Rightarrow x^n=x^{2v}\Rightarrow n=2v \end{align*} $
  • $ \begin{align*} g(x) =\sum_{n=0}^{\infty}\frac{g^{(n)}(0)}{n!}x^n =\sum_{v=0}^\infty\frac{5^v}{v!}x^{2v}\Rightarrow \frac{g^{(n)}(0)}{n!}=\frac{5^v}{v!}\Rightarrow g^{(n)}(0) =\frac{5^vn!}{v!} =\frac{\sqrt{5^n}n!}{\left(\frac{n}{2}\right)!} \end{align*} $

A piecewise definition of the n-th derivative of $g$ would be: $$ g^{(n)}(0)=\begin{cases}\frac{\sqrt{5^n}n!}{\left(\frac{n}{2}\right)!}&n \to even\\ 0&n \to odd\end{cases} $$