Radius of convergence of power series which has factorial term

I am trying to find radius of convergence of the following power series: $\sum_{n\geq 1} n^n z^{n!}$

I tried ratio test but it became complicated, I have never seen such radius of convergence problem with factorial.

• The ratio test can also be applied in this series, using consecutive nonvanishing terms. – hardmath Aug 12 '14 at 13:47

By the Cauchy-Hadamard formula; the radius of convergence of a power series isgiven by $$\frac1{R}=\limsup |a_n|^{1/n}$$

Now , we may plug in and see that $|n^n|^{1/n!} \to 1$ as $n\to \infty$ .So we conclude that $R=1$.

• hmm, I probably miss something, but Cauchy-Hadamard formula requires $a_n = c_n (z-z_0)^n$ format, while we have $n!$. – Alex Aug 12 '14 at 12:20
• @Alex The coefficients are $0$ for all $n$ that aren't factorials. – Daniel Fischer Aug 12 '14 at 12:21
• @DanielFischer $a_n(z)=n^nz^{n!}$, and we want find the value of $z$ for which the series $\sum a_n(z)$ converge, in this case $\sqrt[n]{|a_n(z)|}\neq |n|^{\frac{1}{n!}}$. – Hamou Aug 12 '14 at 12:47
• @Hamou For the Cauchy-Hadamard formula, you consider only the coefficients $a_n$ of the power series $$\sum_{n=0}^\infty a_n (z-z_0)^n.$$ Here, we have $$a_n = \begin{cases} k^k &, n = k!\\ 0 &, (\forall k)(n \neq k!).\end{cases}$$ And thus $$\limsup_{n\to\infty} \lvert a_n\rvert^{1/n} = \lim_{k\to\infty} \lvert k^k\rvert^{1/k!} = \lim_{k\to\infty} k^{1/(k-1)!} = 1.$$ – Daniel Fischer Aug 12 '14 at 12:52
• @DanielFischer: you should have put this as an answer! – Alex Aug 12 '14 at 14:50

By Cauchy's criterion we have $R=1$. EDIT $\alpha_n=\sqrt[n]{n^n|z|^{n!}}=n|z|^{(n-1)!}$, if $|z|>1$ it is clear that $\alpha_n\to \infty$ (the series diverge). If $|z|<1$, then $\alpha_n=\left[\dfrac{n}{(n-1)!}\right]\left[(n-1)!|z|^{(n-1)!}\right]\to 0$ (the series converge).

• you mean not by ratio test or root test? – user169306 Aug 12 '14 at 11:14
• Sorry, root test, but by French it called "Critère de Cauchy", and I am not very good in English. Any correction is welcome – Hamou Aug 12 '14 at 11:19
• The ratio test is also known as Cauchy's ratio test, so additional information is necessary to disambiguate the meaning. – hardmath Aug 12 '14 at 11:36
• The last answer is edited – Hamou Aug 12 '14 at 11:56