An Entry of Ramanujan on Hypergeometric Series in his Notebooks

Example 7, after Entry 43, in Chapter XII of the first Notebook of Srinivasa Ramanujan is proved and, more generally, a summation theorem for ${}_3{F}_2(a,a,x;1+a,1+a+N;1)$, where $N$ is a non-negative integer, is derived.