Hello there, I was having some trouble with this problem:
Find the Taylor series for f centered at 4 if:
$\displaystyle \[f^n(0) = \frac{(-1)^nn!}{3^n(n+1)}\]$
What is the radius of convergence of the Taylor series?
Help appreciates.
Hello there, I was having some trouble with this problem:
Find the Taylor series for f centered at 4 if:
$\displaystyle \[f^n(0) = \frac{(-1)^nn!}{3^n(n+1)}\]$
What is the radius of convergence of the Taylor series?
Help appreciates.
The Taylor expansion of f(*) around z=0 is...
$\displaystyle f(z)= \sum_{n=0}^{\infty} \frac{(-1)^{n}\ z^{n}}{3^{n}\ (n+1)}$ (1)
... and the series (1) has radius of convergence 3 because the singularity of the function in z=-3. Therefore the series of f(*) centered in z=4 has radius of convergence 7...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
The problem is ill defined, we have no sufficient information about $\displaystyle f$ . For example
$\displaystyle f(x)=\begin{Bmatrix} \displaystyle \sum_{n=0}^{\infty}\dfrac{(-1)^n}{3^n(n+1)}x^n& \mbox{if}& |x|<3\\x-4 & \mbox{if}& |x|\geq 3\end{matrix}\quad g(x)=\begin{Bmatrix} \displaystyle \sum_{n=0}^{\infty}\dfrac{(-1)^n}{3^n(n+1)}x^n& \mbox{if}& |x|<3\\|x-4| & \mbox{if}& |x|\geq 3\end{matrix}$
We have $\displaystyle f^{(n)}(0)=g^{(n)}(0)=\dfrac{(-1)^nn!}{3^n(n+1)}$ . However: a) The Taylor series of $\displaystyle f$ centered at $\displaystyle 4$ has radius of convergence $\displaystyle +\infty$ . b) The Taylor series of $\displaystyle g$ centered at $\displaystyle 4$ does not exist .
May be that a further explanation from me is necessary... if we consider the series expansion...
$\displaystyle g(x)= \sum_{n=0}^{\infty} \frac{(-1)^{n}\ x^{n}}{3^{n}}= \frac{1}{1+\frac{x}{3}}$ (1)
...it is not too difficult to obtain...
$\displaystyle f(x)= \sum_{n=0}^{\infty} \frac{(-1)^{n}\ x^{n}}{3^{n}\ (n+1)}= \frac{1}{x}\ \int_{0}^{x} g(\xi)\ \d \xi = \frac{3\ \ln (1+\frac{x}{3})}{x}$ (2)
Now the f(*) defined in (2) is analytic for $\displaystyle x>-3$ and that means that its series expansion centered in $\displaystyle x_{0}>-3$ has radius of convergence $\displaystyle x_{0}+3$ ...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
Well, independently of your explanation I would insist that the only solution to the problem is: The radius of convergence of the Taylor series for $\displaystyle f$ centered at $\displaystyle x_0=4$ (if exists) can be any $\displaystyle \rho\in[0,+\infty]$ depending on the expression of $\displaystyle f^{(n)}(4)$ .