I'm reading Casella's and Berger's Statistical Inference. On page 239 they gives a claim that $$\sum_{x=0}^{330}\binom{300+x-1}x\left(\frac{1}{2}\right )^{300}\left (\frac{1}{2}\right )^x\approx 0.8916.$$ Is that correct? I tried with Wolfram Alpha which gives
sum(binom(299+x,x)*1/2^(300+x),x=0,330)
0.8916...
But Sage gives
sage: x=var('x');N(sum(binomial(299+x,x)/2^(300+x), x, 0, 331),50)
0.89866907836988
Is there a (nice) way to find out which approximation is better?