Generalization of Ramanujan's formula for sums of half-integer powers of consecutive integers via formal Bernoulli series

Faulhaber’s formula expresses the sum of the first $n$ positive integers, each raised to an integer power $p\ge 0$, as a polynomial in $n$ of degree $p+1$. Ramanujan expressed this sum for $p\in \{\frac{1}{2},\frac{3}{2},\frac{5}{2},\frac{7}{2}\}$ as the sum of a polynomial in $\sqrt{n}$ and a certain infinite series. In the present work, we explore the connection to Bernoulli polynomials, and by generalizing those to formal series, we extend the Ramanujan result to all positive half-integers $p$.