Evaluate the infinite sum S=∑n=0∞(2nn)216n(n+1)S = \sum_{n=0}^{\infty} \frac{\binom{2n}{n}^2}{16^n (n+1)}S=∑n=0∞16n(n+1)(n2n)2 using Gauss's hypergeometric theorem.
4π\frac{4}{\pi}π4
2π\frac{2}{\pi}π2
π4\frac{\pi}{4}4π
8π2\frac{8}{\pi^2}π28