Best Answer:
That's an interesting question for which I have just posted a detailed answer at the first link below. I use θ to denote the variable angle of the pendulum and A for its amplitude (the so-called "release angle") called θ_0 in the quoted Wikipedia article.

Here's a summary:

From the normalized nonlinear equation θ'' + sin θ = 0

I derive (essentially in 3 lines) the following relation:

( dx / dt )^2 = (1 - x^2) (1 - k^2 x^2)

by defining k = sin(A/2) and x = sin(θ/2) / k

In that new differential equation, the variables x and t can be separated to obtain a solution "by quadrature", giving t as the integral of a function of x, parametrized by k. (That's a traditional notation; k [not k^2, please!] is called the "modulus" of the relevant animal, called an "elliptic integral", which cannot be expressed with elementary functions.)

In particular, as x goes from 0 to 1 (i.e., θ goes from 0 to A) the time t increases by one fourth of a period. So, the period is proportional to the whole integral from 0 to 1, which goes by the fancy name of "complete elliptic integral of the first kind", denoted by K(k). More precisely, if we drop the above "normalization" convention, we obtain:

T = 2π√(L/g) [ K(k) / (π/2) ]

I prefer not to "simplify" that relation, which is best kept in the form T = T0 [...].

K(k) has a well-known expansion in terms of (even) powers of k from which you can derive the expression of K(sin A/2) as a power series of A. This is the expansion you are quoting from Wikipedia. The rational coefficients do not obey any simple pattern (I give that series explicitly up to its A^20 term and it's a monstrosity).

COMPLEMENT:

I also found something wonderful (which few authors had noticed, as it turns out) by recalling the small article I wrote more than two years ago (second link below) about the connection between K(k) and the classical "arithmetic-geometric mean" (abbreviated AGM and defined in the article):

The true period T of the oscillations of amplitude A in a pendulum whose small-swing oscillations have period T0 is simply given by the relation:

T = T0 / agm ( 1 , cos A/2)

This superb formula is simple, robust and convenient. It allows you to quickly compute T/T0 on a pocket calculator using just a few additions, multiplications and square roots! For example, for an amplitude A of 30 degrees (π/6) the true period of the pendulum is:

T = 1.0174087976 T0

To obtain that many (correct) digits, you'd need to expand the aforementionedl power series of A up to the (monstrous) A^12 term. Things are even worse for larger amplitudes...

Even without the benefit of the above elegant formula, people who have published about the "nonlinear pendulum" have sometimes embarrassed themselves by ignoring the fact that T clearly tends to infinity when the release angle tends to 180 degrees (a rigid pendulum starting from its unstable position of equilibrium directly above its axis of rotation).

A table is provided in my article which gives the ratio of T/T0 for various values of A from 0 to 180 degrees, especially near both extremities of that interval.

Enjoy.

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