Series related to the arcsin
Consequences of the series expansion of 1√1−x vol II
We continue developing some results of series involving the binomial coefficient. This time we show the proof of the following beautiful result posted by
@seriesbot_q:
∞∑n=0(2nn)24n(2n+1)2=√32(1+122−142−152+172+182−...)
Proof:
From the
basic series
1√1−x=∞∑n=0(2nn)22nxn=∞∑n=0(2n−1)!!(2n)!!xn
If we make the transformation
x↦x2
1√1−x2=∞∑n=0(2nn)22nx2n
Integrating from
0 to
t:
∫t01√1−x2dx=arcsin(t)=∞∑n=0(2nn)22n(2n+1)t2n+1
Dividing by
t:
arcsin(t)t=∞∑n=0(2nn)22n(2n+1)t2n
Integrating again from
0 to
12:
∫120arcsin(t)tdt=∞∑n=0(2nn)22n(2n+1)∫120t2ndt=12∞∑n=0(2nn)24n(2n+1)2
⟹∞∑n=0(2nn)24n(2n+1)2=2∫120arcsin(t)tdt
The left hand side is the series that we are looking for. Despite the fact that the integral
∫arcsinxxdx cannot be expressed as a finite combination of elementary functions, we can express it in the form of a new infinite series.
First we have to make a change of variable:
∫120arcsin(t)tdt=∫π60wcos(w)sin(w)dw=∫π60wcot(w)dw(w↦arcsin(t))
Now, recall the following series:
cot(w)=lim
This proof of this result is easy: Consider the "discrete" Laplace transform:
\sum_{k=1}^{\infty} e^{-kt}\sin(kw) = \frac{1}{2} \frac{\sin w}{\cosh t - \cos w} \quad t>0
This follows from:
\begin{align*}
\sum_{k=1}^{\infty} e^{-kt}\sin(kw) =& \sum_{k=0}^{\infty} e^{-kt}\left(\frac{e^{kiw}-e^{-kiw}}{2i}\right)\\
=& \frac{1}{2i} \sum_{k=0}^{\infty} e^{-kt}\left(e^{kiw}-e^{-kiw}\right)\\
=& \frac{1}{2i} \sum_{k=0}^{\infty} e^{k(iw-t)} -\frac{1}{2} \sum_{k=0}^{\infty} e^{k(-t-iw)}\\
=& \frac{1}{2i} \frac{1}{1-e^{iw-t}}- \frac{1}{2i} \frac{1}{1-e^{-t-iw}} \\
=& \frac{1}{2i} \left(\frac{e^t}{e^t-e^{iw}}- \frac{e^t}{e^t-e^{-iw}} \right)\\
=& \frac{1}{2i} \left(\frac{e^{t+iw}-e^{t-iw}}{e^{2t}-e^{t-iw}-e^{t+iw}+1}\right)\\
=& \frac{1}{2i} \left(\frac{e^{iw}-e^{-iw}}{e^{t}+e^{-t}-(e^{-iw}+e^{iw})}\right)\\
=& \frac{1}{2} \frac{\frac{e^{iw}-e^{-iw}}{2i}}{\frac{e^{t}+e^{-t}-(e^{-iw}+e^{iw})}{2}}\\
=& \frac{1}{2} \frac{\sin w}{\cosh t - \cos w}\\
\end{align*}
If we take the limit as
\displaystyle t \to 0+
\lim_{t \to 0+ }\sum_{k=1}^{\infty}e^{tk} \sin(kw) = \frac{1}{2} \frac{\sin w}{1 - \cos w}
and using the following identities
\sin\left(\theta\right) = 2\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right)
1-\cos(\theta) = 2\sin^2\left(\frac{\theta}{2}\right)
we have
\lim_{t \to 0+} \sum_{k=1}^{\infty} e^{-tk}\sin(kw) = \frac{1}{2} \frac{\sin w}{1 - \cos w} = \frac{1}{2}\cot\left(\frac{w}{2}\right)
therefore
\cot\left(w\right) = \lim_{t \to 0+} 2\sum_{k=1}^{\infty} e^{-tk}\sin(2kw) \tag{2}
This seems to imply that
\cot\left(w\right) \stackrel{?}{=} 2\sum_{k=1}^{\infty} \sin(2kw)
However, this is not true... at least formally: note that the series
\displaystyle \sum_{k=1}^{ \infty}\sin(2kw) does not converge.
But we can calculate the integral taking the limit as
t \to 0+:
\begin{align*}
I = \int_{0}^{\frac{1}{2}}\frac{\arcsin(t)}{t}dt =& \int_{0}^{\frac{\pi}{6}} \frac{w\cos(w)}{\sin(w)} dw\\
=& \int_{0}^{\frac{\pi}{6}} w\cot(w) dw\\
=& 2\int_{0}^{\frac{\pi}{6}} w\left[\lim_{t \to 0+} \sum_{k=1}^{\infty}e^{-tk}\sin(2kw)\right] dw \quad (\textrm{from (2)})\\
=& \lim_{t \to 0+} 2\sum_{k=1}^{\infty} e^{-tk}\int_{0}^{\frac{\pi}{6}} w\sin(2kw) dw\\
=& \lim_{t \to 0+} \frac{1}{2}\sum_{k=1}^{\infty}e^{-tk} \frac{1}{k^2}\int_{0}^{\frac{\pi k}{3}} s\sin(s) dw \quad (s \mapsto 2kw)\\
\stackrel{IBP}{=}& \lim_{t \to 0+} \sum_{k=1}^{\infty}e^{-tk}\frac{1}{2k^2} \left[-\frac{\pi k}{3}\cos\left(\frac{k\pi}{3}\right) + \int_{0}^{\frac{\pi k}{3}} \cos sds \right]\\
=& \lim_{t \to 0+} \sum_{k=1}^{\infty}e^{-tk}\frac{1}{2k^2} \left[\sin\left(\frac{\pi k}{3}\right)-\frac{\pi k}{3}\cos\left(\frac{k\pi}{3}\right) \right]\\
=& \lim_{t \to 0+} \frac{1}{2} \sum_{k=1}^{\infty} e^{-tk}\frac{\sin\left(\frac{\pi k}{3}\right)}{k^2}-\lim_{t \to 0} \frac{\pi}{6}\sum_{k=1}^{\infty}e^{-tk}\frac{\cos\left(\frac{k\pi}{3}\right)}{k}\\
=& \frac{1}{2} \sum_{k=1}^{\infty}\frac{\sin\left(\frac{\pi k}{3}\right)}{k^2}- \frac{\pi}{6}\sum_{k=1}^{\infty}\frac{\cos\left(\frac{k\pi}{3}\right)}{k}\\
\end{align*}
where the last equality follows from the fact that the series
\sum_{k=1}^{\infty}\frac{\sin\left(\frac{\pi k}{3}\right)}{k^2} \quad \sum_{k=1}^{\infty}\frac{\cos\left(\frac{k\pi}{3}\right)}{k}
are well known Fourier series that converge.
Now we will prove that the second series is equal to zero:
From Euler's formula we have
\sum_{k=1}^{\infty} \frac{\cos\left(\frac{k\pi}{3}\right)}{k} = \frac{1}{2}\sum_{k=1}^{\infty} \frac{e^{\frac{i\pi k}{3}}}{k}+ \frac{1}{2}\sum_{k=1}^{\infty} \frac{e^{-\frac{i\pi k}{3}}}{k}
We can split the sum into two because the series
\sum_{k=1}^{\infty} \frac{z^k}{k}
converges when
\displaystyle |z|<1. But this series is just the Laurent series for the principal branch of the
-\ln(1-z) function:
\ln(1-z) = -\sum_{k=1}^{\infty} \frac{z^k}{k}
Hence we can write:
\sum_{k=1}^{\infty} \frac{\cos\left(\frac{k\pi}{3}\right)}{k} = \frac{1}{2}\sum_{k=1}^{\infty} \frac{e^{\frac{i\pi k}{3}}}{k}+ \frac{1}{2}\sum_{k=1}^{\infty} \frac{e^{-\frac{i\pi k}{3}}}{k} = -\frac{1}{2}\ln\left(1-e^{\frac{i\pi}{3}}\right)-\frac{1}{2}\ln\left(1-e^{-\frac{i\pi}{3}}\right) = -i\frac{\pi}{6}+ i\frac{\pi}{6} =0
Therefore
I =\int_{0}^{\frac{1}{2}}\frac{\arcsin(t)}{t}dt = \frac{1}{2} \sum_{k=1}^{\infty} \frac{\sin\left(\frac{\pi k}{3}\right)}{k^2}
If we tabulate the sequence
\displaystyle \sin\left(\frac{\pi k}{3}\right) \textrm{ for } k=1,2,3,... we find:
\left(\sin\left(\frac{\pi k}{3}\right)\right)_{k} = \left(\frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{2}, 0 ,-\frac{\sqrt{3}}{2}, -\frac{\sqrt{3}}{2},0, \frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{2},... \right) = \frac{\sqrt{3}}{2}(1,1,0,-1-1,0,1,1,...)
Therefore
I =\int_{0}^{\frac{1}{2}}\frac{\arcsin(t)}{t}dt = \frac{1}{2} \sum_{k=1}^{\infty} \frac{\sin\left(\frac{\pi k}{3}\right)}{k^2}= \frac{\sqrt{3}}{4}\left(1+\frac{1}{2^2}-\frac{1}{4^2}-\frac{1}{5^2} + \frac{1}{7^2}+\frac{1}{8^2}+...\right) \tag{3}
Putting all the ingredients together:
\begin{align*}
\sum_{n=0}^{\infty} \frac{\binom{2n}{n}}{2^{4n}(2n+1)^2} = & 2\int_{0}^{\frac{1}{2}}\frac{\arcsin(t)}{t}dt \quad (\textrm{from (1)}) \\
=& \frac{\sqrt{3}}{2}\left(1+\frac{1}{2^2}-\frac{1}{4^2}-\frac{1}{5^2} + \frac{1}{7^2}+\frac{1}{8^2}+...\right) \quad (\textrm{from (3)})
\end{align*}
Therefore
\boxed{\sum_{n=0}^{\infty} \frac{\binom{2n}{n}}{2^{4n}(2n+1)^2} = \frac{\sqrt{3}}{2}\left(1+\frac{1}{2^2}-\frac{1}{4^2}-\frac{1}{5^2} + \frac{1}{7^2}+\frac{1}{8^2}-...\right) }