# Summation does not simplify

Inspired by the identity listed in this problem, I tried

2 π Sqrt[2]/9801 Sum[((4 k)! (1103 + 36390 k))/((k!)^4  396^(4 k)),
{k, 0, \[Infinity]}]


and got

(1/(418435621956 Sqrt[2])) π
(188362204272 HypergeometricPFQ[{1/4, 1/2, 3/4}, {1, 1}, 1/96059601] +
6065 HypergeometricPFQ[{5/4, 3/2, 7/4}, {2, 2}, 1/96059601])


FullSimplify did not lead to any simplification or the desired answer: $1$.

However, N[%] yielded 1.

Why didn't this summation equality get evaluated symbolically to yield the answer $1$?

• In fact, N[%-1] doesn't yield zero, it yields $8.8\cdot 10^{-9}$. Are you completely sure the sum is indeed one? Or may it just be close? – AccidentalFourierTransform Feb 3 '18 at 0:37
• I'm assuming the equality identity (not approximation) given in the linked problem is correct, but perhaps this is a mistake. – David G. Stork Feb 3 '18 at 0:39
• I got the one when I run this and I ask for the numerical value. However, when I subtract one from the other I get something close to zero, namely $-2.81647*10^{-9}$. – Darth_Bane Feb 3 '18 at 0:42

The correct formula is with a 26390 instead of 36390. See (78) here. Mathematica now outputs the correct result:
\[Pi] Sqrt[8]/9801 Sum[((4 k)! (1103 + 26390 k))/((k!)^4 396^(4 k)), {k, 0, \[Infinity]}]