A typo in a formula of Ramanujan? 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?

The series in question converges slowly, $(8k+1) \left( \frac{(1/4)_k}{k!} \right)^4 \sim \frac{8}{k^2 \Gamma^4(1/4)}$, hence it may be that you have not computed enough terms.

The sum represents a value of a hypergeometric function: $$ \sum_{k=0}^\infty (8k+1) \left( \frac{(1/4)_k}{k!} \right)^4 = {}_4F_3\left( \frac{1}{4},\frac{1}{4}, \frac{1}{4}, \frac{1}{4}; 1,1,1 | 1\right) - \frac{1}{32} {}_4F_3\left( \frac{5}{4},\frac{5}{4}, \frac{5}{4}, \frac{5}{4}; 2,2,2 | 1\right) $$ Evaluating these numerically agrees with the expression in terms of $\Gamma$ constant:


In[18]:= Sum[(8 k + 1) (Pochhammer[1/4, k]/k!)^4, {k, 0, \[Infinity]}]

Out[18]= 1/32 (32 HypergeometricPFQ[{1/4, 1/4, 1/4, 1/4}, {1, 1, 1},
1] + HypergeometricPFQ[{5/4, 5/4, 5/4, 5/4}, {2, 2, 2}, 1])

In[19]:= N[%, 30]

Out[19]= 1.06267989991684365118249019510

In[20]:= N[(2^(3/2)/(Sqrt[Pi] Gamma[3/4]^2)), 30]

Out[20]= 1.06267989991684365118249019510