closed form for the series

Question

I don't remember any method to compute the closed from for the following series. $$ \sum_{k=0}^{\infty}\binom{3k}{k} x^k .$$

I tried by putting $\binom{3k}{k}$ in Mathematica for different $k$ and asking for the generating function it deliver a complicated formula which is the following. $$ \frac{2\cos[\frac{1}{3} \sin^{-1}(\frac{\sqrt{3x}}{2})]}{\sqrt{4-27x}} $$

I was wondering if there is any simple form?

$$\frac{2 \cos \left(\frac{1}{6} \cos ^{-1}\left(1-\frac{27 x}{2}\right)\right)}{\sqrt{4-27 x}}$$ looks slightly simpler. — Claude Leibovici, Oct 27 '16 at 10:00
See https://oeis.org/A005809 for more information and references — Robert Z, Oct 27 '16 at 10:01

DonAntonio · Answer 1 · 2016-10-27T20:04:16.953

3

That's a power series about $\;x_0=0\;$ and whose sequence of coefficients is

$$a_k=\binom{3k}kx^k=\frac{(3k)!}{k!(2k)!}\implies\;\left|\frac{a_{k+1}}{a_k}\right|=\frac{(3k+3)!}{(k+1)!(2k+2)!}\cdot\frac{k!(2k)!}{(3k)!}|x|=$$

$$=\frac{(3k+1)(3k+2)(3k+3)}{(k+1)(2k+1)(2k+2)}|x|\xrightarrow[k\to\infty]{}\frac{27}{4}|x|$$

and thus the series converges for

$$\frac{27}4|x|<1\iff |x|<\frac4{27}$$

edited Oct 27 '16 at 20:04

answered Oct 27 '16 at 09:58

DonAntonio

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What? It converges for all $|x|<\frac{4}{27}$. That's what the ratio test tells you. – Bob Jones Oct 27 '16 at 19:50
Stirling's formula tell you there is convergence when $|x| \lt \frac{4}{27}$. – Ron Gordon Oct 27 '16 at 20:01
@BobJones You're right: I forgot completely the $;x;$ ...:) :) Editing now, thanks. – DonAntonio Oct 27 '16 at 20:03
@GGT Please do read the edition and correction of my answer. – DonAntonio Oct 27 '16 at 20:04
@RonGordon Thanks, but read above: why would I want to work with Stirling if the ratio test is so easy to apply here...when no mistakes are done...? – DonAntonio Oct 27 '16 at 20:05
"when no mistakes are done." You answered the question yourself. – Ron Gordon Oct 27 '16 at 20:06
@RonGordon Oh, but the mistake was there, and I wouldn't have seen it without Bob, and to mend it the ratio test is much quicker, imo. – DonAntonio Oct 27 '16 at 20:07

score 0 · Answer 2 · edited Apr 13 '17 at 12:21

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Hint: A closed form can be found by means of the Lagrange Inversion Formula.

An answer based upon this method is given at this MSE link.

edited Apr 13 '17 at 12:21

Community

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answered Oct 28 '16 at 13:48

Markus Scheuer

108,315

closed form for the series

2 Answers2