Solution
Central Binomial Series
Show the problem again
Evaluate $\sum_{n=0}^{\infty} \binom{2n}{n} / 8^n = 1 + 2/8 + 6/64 + 20/512 + \dots$. Give an exact closed form such as sqrt(5); a decimal correct to three places is also accepted.
Worked solution
The answer is sqrt(2). The central binomial coefficients have generating function $\sum_n \binom{2n}{n} x^n = (1 - 4x)^{-1/2}$ for $|x| < 1/4$. One way to see it: $\binom{-1/2}{n}(-4)^n = \binom{2n}{n}$, so this is just the binomial series for $(1-4x)^{-1/2}$. At $x = 1/8$ the sum is $(1 - 1/2)^{-1/2} = \sqrt{2} \approx 1.4142$. The terms shrink only like $1/\sqrt{\pi n}\,2^{-n}$, so the partial sums creep up on $\sqrt{2}$ quickly but visibly.
Source: Classic generating-function identity in wide circulation; no traceable original source. Statement written for AxiomIQ.