I did two calculations that I think are wrong but I am not sure why.
I have to compute the convergence radius of the following power series
a) $\sum_0^{\infty} \ln(k!)x^k$,
b) $\sum_0^{\infty} k8^kx^{3k}$.
Here's my attempt:
a)
$$ \limsup_{k \rightarrow \infty} \sqrt[k]{a_k} = \limsup_{k \rightarrow \infty} \sqrt[k]{\ln(k!)} = \ln (\limsup_{k \rightarrow \infty} \sqrt[k]{k!})= \infty. $$
Therefore the convergence radius should be $0$ and the series only converges for the value $x=0$. Unfortunately, Wolfram alpha gives me another answer. Where's the mistake ?
b)
$$ \limsup_{k \rightarrow \infty} \sqrt[k]{a_k} = \limsup_{k \rightarrow \infty} \sqrt[k]{k8^k} = \limsup_{k \rightarrow \infty} \sqrt[k]{k} \sqrt[k]{8^k} = 8 \limsup_{k \rightarrow \infty} \sqrt[k]{k} = 8. $$
I would now conclude that the convergence radius is $\frac{1}{8}$, but it appears to be $\frac{1}{2}$, what did I miss ?
Thank you for your help.
Answer
For a) you have made the error of assume the log and the exponent commute. That is $$\ln \sqrt[k]{k!} \ne \sqrt[k]{ \ln k!}$$ and hence $$ \limsup_{k \to \infty} \sqrt[k]{ \ln k!} \ne \ln \left( \limsup_{n \to \infty} \sqrt[k]{k!} \right)$$
I have to disagree with the other answer on how to evaluate this limit (after the error). Using Stirling's approximation gives $$ \sqrt[k]{k!} \sim \sqrt[k]{ \sqrt{2 \pi k} \left( \frac{k}{e} \right)^k } = \left(2 \pi k \right)^{\frac{1}{2k}} \frac{k}{e} \to \infty$$
Of course the actual problem was the one I pointed out before. Here's how I would attack it.
$$\lim_{k \to \infty} \sqrt[k]{ \ln k!}=\lim_{k \to \infty} \frac{ \ln (k+1)!}{\ln (k!)}=\lim_{k \to \infty} \frac{\sum_{j=0}^{k+1} \ln j}{\sum_{j=0}^{k} \ln j}=\lim_{k \to \infty} \frac{\ln (k+1)}{\ln k}=1$$ Note I used the Stolz–Cesàro theorem in the third equality.
I agree with the other answer for (b). You missed the $3k$ in the exponent
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