I was looking at Wikipedia's page for nth root and there is a section given for representation of nth root with infinite series. I was trying to calculate the $$ \sqrt[3]{12} $$ with$$ (1+x)^{s/t}=\sum_{n=0}^\infty {\frac {\prod_{k=0}^{n-1} (s-kt)} {n!t^n} x^n}$$ and compare the result with results given by nth root algorithm. If I am correct the infinite serie would become something like this $$ (1+11)^{1/3}=\sum_{n=0}^\infty {\frac {\prod_{k=0}^{n-1} (1-3k)} {n!3^n} 11^n}$$ I was expecting a number close to 2.3 but it gave me a really large number for at least 7 steps. a sample of my steps
I don't know why this happened. I think that the way I am calculating might be wrong because I had to calculate each product manually for each step maybe.if this is the case I would like to know what were my mistakes.
footnote : My knowledge of math is quite limited.I also couldn't find any source for this infinite serie.