Ramanujan summation

Summary

Ramanujan summation assigns a value to a series $\sum_{n \ge 1} f(n)$ by taking the constant term in the Euler–Maclaurin formula for its partial sums. The value is expressed through $f(0)$, the odd derivatives of $f$ at $0$ and the Bernoulli numbers. It gives $1 + 2 + 3 + \cdots = -\frac{1}{12}$ and $1 + 1 + 1 + \cdots = -\frac12$, the values $\zeta(-1)$ and $\zeta(0)$ of the Riemann zeta function. Unlike most summability methods, it is neither regular nor stable under shifting the index.