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Is the following series convergent or divergent? $$\sum_{n=1}^\infty \frac{3^n n^2}{n!}$$

I used root test to check but I am stuck at

$$\lim\limits_{n \to \infty} \sqrt[n]\frac{3^n n^2}{n!} = \lim\limits_{n \to \infty} \frac{3n^\frac{2}{n}}{n!^\frac{1}{n}}$$

I don't know how to calculate $\lim\limits_{n \to \infty} n!^\frac{1}{n}$. So I'm guessing maybe I am going at it in the wrong way.

vitamin d
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Yahya Fati
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  • The limit $\lim\limits_{n \to \infty} n!^\frac{1}{n}$ diverges to infinity. See Stirling. – WhatsUp Jun 02 '21 at 14:47
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    Using the ratio test is simpler. – Bernard Jun 02 '21 at 14:48
  • There is a result that $\lim\limits_{n\to\infty}(a_n)^{\frac{1}{n}} = \lim\limits_{n\to \infty} \frac{a_{n+1}}{a_n}$, provided the latter limit exists (assuming $a_n$ to be positive). Can you use it? – Yathi Jun 02 '21 at 14:49
  • If everything isn't raised to the $n$ generally you don't want to use the root test. However, if you really wanted to, you could use Stirling's approximation https://en.wikipedia.org/wiki/Stirling%27s_approximation to convert $n!$ into something you can take the root of easier – Alan Jun 02 '21 at 14:51
  • Also $\sum_{n=1}^\infty \frac{z^n n^2}{n!} = e^z z (z + 1)$ – lhf Jun 02 '21 at 15:00
  • https://math.stackexchange.com/questions/136626/lim-limits-n-to-infty-sqrtnn-is-infinite – Hans Lundmark Jun 02 '21 at 15:05

2 Answers2

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Checking convergence without applying a specific convergence test: $$\sum_{n=1}^\infty \frac{3^n n^2}{n!}\leqslant\sum_{n=1}^\infty \frac{3^n\cdot3^n}{n!}=\sum_{n=1}^\infty \frac{9^n}{n!}=e^9-1<\infty$$ Also one can show with Stirlings formula that the limit $\lim\limits_{n \to \infty} n!^\frac{1}{n}$ does not exist.

vitamin d
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I would suggest using the ratio test: If $\lim_{n\to\infty}\left |\frac{a_{n+1}}{a_n}\right | < 1$ we have an absolutely convergent (and in this case also convergent) series:

$$ \frac{\frac{3^{n+1}(n+1)^2}{(n+1)!}}{\frac{3^nn^2}{n!}} = \frac{3(n+1)^2}{n^2(n+1)} = \frac{3n+3}{n^2} \overset{n\to\infty}{\longrightarrow} 0 $$

Edit: How to calculate the root-check: \begin{align} \lim_{n\to\infty}\frac{3n^\frac{2}{n}}{n!^\frac 1 n} &= 3\lim_{n\to\infty}\sqrt[n]{\frac{n^2}{n!}} = 3\lim_{n\to\infty}\sqrt[n]{\frac{n}{(n-1)!}}\leq 3\lim_{n\to\infty}\sqrt[n]{\frac{n}{n(n-3)!}} \\ &= 3\lim_{n\to\infty}\sqrt[n]{\frac{1}{(n-3)!}} < 1 \end{align} The inequality is correct because $(n-1)! > n(n-3)! \Leftrightarrow (n-1)(n-2) > n$ is true for $n\geq 4$. As mentioned in above comments $\sqrt[n]{n!}\to\infty$ so the term in the limit $\to 0$, it doesn't matter that a finite amount (three) of factors $n,n-1,n-2$ are missing.

We don't need to specify the exact limit, we only need to know if it is $<1$, which is shown above.

LegNaiB
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