TITLE: A typo in a formula of Ramanujan?
QUESTION [6 upvotes]: In Mathworld's article Gamma function, in line (96), we find the formula,
$\sum_{k=0}^\infty (8k+1)\left(\frac{\Gamma(k+\frac{1}{4})}{k!\;\Gamma(\frac{1}{4})}\right)^4 = 2^{3/2}\frac{1}{\sqrt{\pi}\,\left(\Gamma(3/4)\right)^2}$
On a whim, I evaluated the LHS and RHS using Mathematica to 100-digit precision, and found the first few digits as,
$\text{LHS} = 1.062679901\dots$
$\text{RHS} = 1.062679899\dots$
Ahem, they don't match.  If it is a typo, then I find it interesting it is exceedingly close.
So what is the problem? 1) Did I input it in Mathematica wrongly? 2) Is there a typo, or misplaced symbol by authors after Ramanujan (Weisstein gives Hardy et al as references) 3) Or was Ramanujan just mistaken?

REPLY [3 votes]: FWIW, Maple 16 gets the sum right (symbolically, and numerically to within a reasonable roundoff error).

L:= Sum((8*k+1)*(GAMMA(k+1/4)/k!/GAMMA(1/4))^4,k=0..infinity);
R:= value(L);

$$ R := 2\,{\frac {\sqrt {2}}{\sqrt {\pi } \left( \Gamma  \left( 3/4 \right) 
 \right) ^{2}}}$$

evalf(L - R, 100);

$$ 0.1\ 10^{-98} $$