Does series $4^{n}×\frac{(n!)^2}{(2n)!}$ converge or diverge?

Question

I know $4^{n}$ diverges, and $\frac{(n!)^2}{(2n)!}$ converges. I also think that their product diverges since the term of this series is increasing but I don't know how to prove it. I tried ratio test which is inconclusive, and some other tests.

Answer 1

Let $a_n:=4^n\cdot\frac{(n!)^2}{(2n)!}$, and note that $2^n\cdot n!=2\cdot4\cdot6\cdots 2n$. Hence

$$a_n=4^n \frac{(n!)^2}{(2n)!}=\prod_{k=1}^{n}\frac{2k}{2k-1}=\exp\left(\sum_{k=1}^n\log\left(\frac{2k}{2k-1}\right)\right)\ge \exp\left(n\log\left(\frac{2n}{2n-1}\right)\right)\tag1$$

and

$$\lim_{x\to\infty}x\log\left(\frac{2x}{2x-1}\right)=\lim_{x\to\infty}\frac{\frac{2(2x-1)-4x}{(2x-1)^2}}{-\left(\frac{2x}{2x-1}\right)\frac1{x^2}}=\lim_{x\to\infty}\frac{2x^2}{(2x-1)2x}=\frac12\tag2$$

Thus $\lim a_n\ge \sqrt e$, so $\sum a_n=\infty$.

---

A more simple approach is expanding and chopping the product on $(1)$, that is

$$\prod_{k=1}^n\frac{2k}{2k-1}=\prod_{k=1}^n\left(1+\frac1{2k-1}\right)=1+\sum_{k=1}^n\frac1{2k-1}+\ldots\tag3$$

Then we have the lower bound

$$a_n\ge 1+\sum_{k=1}^n\frac1{2k-1}=1+\frac12\sum_{k=1}^n\frac1{k-1/2}\ge 1+\frac12\sum_{k=1}^n\frac1k\tag4$$

Hence $\lim a_n=\infty$ (because $\sum\frac1k=\infty$).

Answer 2

Stirling's approximation? Naah, let us go for a greater overkill. Since the series $$ \sum_{n\geq 0}\left[\frac{1}{4^n}\binom{2n}{n}\right]^3 $$ is convergent to $\frac{\pi}{\Gamma\left(\frac{3}{4}\right)^4}$ due to the relation with the squared complete elliptic integral of the first kind (identity $(7)$ at $k=\frac{1}{\sqrt{2}}$), its main term is convergent to zero and your sequence is divergent.

---

Seriously, an elementary approach. Since $2\cos\theta=e^{i\theta}+e^{-i\theta}$ and $\int_{0}^{2\pi}e^{ni\theta}e^{-mi\theta}\,d\theta = 2\pi\delta(m,n)$, $$ \frac{1}{4^n}\binom{2n}{n}=\frac{2}{\pi}\int_{0}^{\pi/2}\cos^{2n}\theta\,d\theta. $$ By the dominated/monotone convergence theorem, the limit of both sides as $n\to +\infty$ is zero, hence your sequence is divergent. We also have that $\left\{\frac{1}{4^n}\binom{2n}{n}\right\}_{n\geq 1}$ is log-convex due to the Cauchy-Schwarz inequality and the previous integral representation.

---

Yet another elementary approach. You may prove in a combinatorial fashion that $$ \sum_{k=0}^{n}\binom{2k}{k}\binom{2n-2k}{n-k} = 4^n \tag{Convolution}$$ hence it follows that $$ \left[\sum_{k=0}^{n}\frac{1}{4^k}\binom{2k}{k}\right]^2 \leq \sum_{k=0}^{2n} 1 = 2n $$ and $$ \sum_{k=0}^{n}\frac{1}{4^k}\binom{2k}{k} \leq \sqrt{2n}. $$ On the other hand we also have

$$ \left[\sum_{k=0}^{n}\frac{1}{4^k}\binom{2k}{k}\right]^2 \geq \sum_{k=0}^{n} 1 = n $$ hence $$ \sqrt{n}\leq \sum_{k=0}^{n}\frac{1}{4^k}\binom{2k}{k} \leq \sqrt{2n}.\tag{SumInequality} $$ Since the sequence $\left\{\frac{1}{4^n}\binom{2n}{n}\right\}_{n\geq 1}$ is decreasing, the previous inequality implies that $\frac{1}{4^n}\binom{2n}{n}\ll\frac{1}{\sqrt{n}}$ as $n\to +\infty$. The correct asymptotic behaviour is given by Wallis' product and it is $$ \frac{1}{4^n}\binom{2n}{n}\sim\frac{1}{\sqrt{\pi n}}.\tag{GoodToKnow}$$

Answer 3

By Stirling's approximation

$$n! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n$$

we have

$$4^n \frac{(n!)^2}{(2n)!}\sim 4^n \frac{2 \pi n\left(\frac{n}{e}\right)^{2n}}{\sqrt{4 \pi n}\left(\frac{2n}{e}\right)^{2n}}=\sqrt{\pi n}\to \infty$$

then the series diverges.