.

A related function is the alternating zeta function, also known as the Dirichlet eta function:

.

Both series are clearly divergent for .

Analytic continuation of the zeta function over the whole complex plane implies the functional relations:

and

.

For or a negative integer, the following absurd results are implied:

,

,

,

and so on. However, in certain specialized mathematical contexts, finite values can be assigned to such divergent series. Ramanujan was a famous proponent of such series. The sum of the natural numbers given by the last formula is, in fact, known as the Ramanujan summation. The subject received a burst of popular interest in 2014 as a result of a YouTube video [1] and an article in The New York Times [2]. The formula turns out to be very significant in dealing with infinities in string theory [3] and also in the theory of the Casimir effect for the vacuum energy of the electromagnetic field.

This Demonstration shows the results of a number of divergent series represented by zeta and eta functions with negative arguments, with , , , and . We also consider the case , which does not give a simple fraction.