#

## Frauds

Gregory's series
for , truncated at **500,000** terms gives to forty places

Only the underlined digits are wrong. This is explained by the following Theorem.

## Theorem.

* For integer ***N** divisible by **4** the following
asymptotic expansion holds:
where the coefficients are
the even Euler numbers **1**, **-1**, **5**, **-61**, **1385**,
.
Gregory's series requires more terms than there are particles
in the universe to compute 100 digits of .

However, with **N = 200,000** and correcting using
the first thousand even Euler numbers gives over **5,000** digits of .

See * Pi, Euler numbers and asymptotic expansions* by J. Borwein, P. Borwein & K. Dilcher
in the MAA Monthly ** 96** (1989) 681-687.

## Excessive Fraud

## Sum.

(correct to over **42** billion digits)

The sum arises from an application of Poisson
summation or equivalently as a modular transformation of a theta function.

## Conjecture.

No one will ever know the th digit of
(or the th).