We will see how to accelerate the convergence of an alternating series to prove the above formula for
Start with
and write it as
by distributing half of each term to two adjacent terms.
For example applying this transformation to
we get
Applying it again, we get
After few more times we get,
In general,
For general sequence the Euler tranform maps the sequence to
We see that by repeating the process shown in the example that
Riemann Zeta Formula:
We apply this Euler’s transformation for
to get
This expression can be seen to be defined for all values although the initial expression only valid for
Using this formula we can quickly compute the following