Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

$$ \sum^{\infty}_{j=0} \frac{(j!)^2}{(2j)!} = \frac{2 \pi \sqrt{3}}{27}+\frac{4}{3} $$

The above series is in a homework sheet. We're not expected to find the limit, just prove its convergence. That's easy, but since we were given the limit, it got me thinking about how to find such a limit.

If anyone could point me in the right direction, I'd be happy to discover it on my own, but after a few hours of searching, I don't feel much closer.

share|improve this question
    
I think that this will be related to beta functions –  Amr Mar 3 '13 at 13:21

4 Answers 4

up vote 10 down vote accepted

Trying to give you something simpler...

Euler found following development for $\arcsin(x)$ : $$f(x):=2\arcsin(x)^2=\sum_{n=1}^\infty \frac{(2x)^{2n}}{n^2\binom{2n}{n}}$$ with $\displaystyle\binom{2n}{n}=\frac{(2n)!}{(n!)^2}$ a central binomial coefficient (see too $(8)$ there).

Compute the derivative $f'(x)$ to get : $$f'(x):=4\frac{\arcsin(x)}{\sqrt{1-x^2}}=4\sum_{n=1}^\infty \frac{(2x)^{2n-1}}{n\binom{2n}{n}}$$ Multiply by $x$ and compute the derivative again : $$(xf'(x))':=4\frac{\arcsin(x)}{\sqrt{1-x^2}}-4\frac x{x^2-1}+4\frac{x^2\arcsin(x)}{\sqrt{1-x^2}^3}=8\sum_{n=1}^\infty \frac{(2x)^{2n-1}}{\binom{2n}{n}}$$

replace $x$ by $\frac 12$ (so that $2x=1$ at the right) to get (dividing by $8$) : $$\frac{4\sqrt{3}\,\arcsin\left(\frac 12\right)}{9}+\frac 13=\sum_{n=1}^\infty \frac 1{\binom{2n}{n}}$$ Use $\arcsin(1/2)=\frac{\pi}6$ and add the $n=0$ term at the right ($1$) to conclude that : $$\boxed{\displaystyle\sum_{n=0}^\infty \frac 1{\binom{2n}{n}}=\frac{2\pi\sqrt{3}}{27}+\frac 43}$$

share|improve this answer
    
(+1) Ingenious! –  L. F. Mar 3 '13 at 14:58
    
Thanks @L.F. ${}$! –  Raymond Manzoni Mar 3 '13 at 15:20

Notice that: $$ \frac{j!^2}{(2j)!} = (2j+1) \operatorname{Beta}(j+1,j+1) = (2j+1) \int_0^1 t^j (1-t)^j \mathrm{d}t $$ Now, interchanging the summation and the integration: $$ \sum_{j=0}^\infty \frac{j!^2}{(2j)!} = \int_0^1 \sum_{j=0}^\infty (2j+1)(t(1-t))^j \mathrm{d} t = \int_0^1 \frac{1+t(1-t)}{(1-t(1-t))^2} \mathrm{d} t $$ The latter integral is evaluated integrating by parts and reducing it to the table integral: $$ \int_0^1 \frac{1+t(1-t)}{(1-t(1-t))^2} \mathrm{d} t = \left[ \frac{2}{3} \frac{2t-1}{1-t+t^2} + \frac{2}{3 \sqrt{3}} \arctan\left(\frac{2t-1}{\sqrt{3}} \right) \right]_{t=0}^{t=1} =2 \left(\frac{2}{3} + \frac{\pi}{9 \sqrt{3}}\right) $$ This establishes the result.

Alternatively, you could note that the summand $c_j$ is a hypergeometric term, which means that the ratio of successive terms is a rational function of $j$ $$ \frac{c_{j+1}}{c_j} = \frac{j+1}{4j+2} = \frac{(j+1)(j+1)}{j+\frac{1}{2}} \frac{1}{4} \frac{1}{j+1} $$ which means that the sum corresponds to the defining series of the Gauss's hypergeometric function: $$\begin{eqnarray} \sum_{j=0}^\infty \frac{j! \cdot j!}{(2j)!} &=& \sum_{j=0}^\infty \frac{(1)_j (1)_j}{\left(\frac{1}{2}\right)_j} \frac{(1/4)^j}{j!} = {}_2F_1\left(\left.\begin{array}{cc} 1 & 1\\ & \frac{1}{2} \end{array} \right| \frac{1}{4}\right) \\ &=& \left.\frac{1}{1-z} \left(1 + \sqrt{\frac{z}{1-z}} \arcsin\sqrt{z}\right)\right|_{z=1/4} = \frac{4}{3} + \frac{2 \pi}{9 \sqrt{3}} \end{eqnarray} $$

share|improve this answer

Here's an alternative method I read from somewhere (Can't remember the source):

We compute the generating function $f(x)$ of $a_j=\frac{4^j}{\binom{2j}{j}}$.

Note that $a_0=1, a_1=2$, and $(2j+1)a_{j+1}=2(j+1)a_j$, so $2(j+1)a_{j+1}-a_{j+1}=2ja_j+2a_j$.

Thus $2f'(x)-\frac{f(x)-1}{x}=2xf'(x)+2f(x)$, so $2x(1-x)f'(x)-(1+2x)f(x)+1=0$

Solving the differential equation with the constraint $f(0)=a_0=1, f'(0)=a_1=2$, we get $f(x)=\frac{1}{1-x}(\sqrt{\frac{x}{1-x}}\arctan{\sqrt{\frac{x}{1-x}}}+1)$.

To get the required sum, we now substitute $x=\frac{1}{4}$, giving $\frac{2\pi\sqrt{3}}{27}+\frac{4}{3}$.

share|improve this answer

Ok, so $$\int_0^1x^k(1-x)^kdx=\dfrac{(k!)^2}{(2k+1)!}$$ For a proof refer to,Good ways to integrate $\int_0^1 x^{k+1} (1-x)^k dx$?

Let $y= 1-x$ $$\sum_{k=0}^{\infty}=\dfrac{(k!)^2}{(2k)!}=\sum_{k=0}^\infty(2k+1)\int_0^1(xy)^kdx=\int_0^1\dfrac{1}{1-xy}dx+\int_0^1\dfrac{2xy}{(1-xy)^2}dx$$ This is by interchanging order of summation and integration and then using formula for the geometric series and a related series. Evaluating these we get the answer.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.