I don't think wolframalpha says it is divergent: https://www.wolframalpha.com/input?i=sum%28binomial%28-2%2F5%2C+n%29%2C+%28n%2C+1%2C+inf%29%29
It does not compute the sum in closed form but gives a formula for the partial sums as 2^(-2/5)-1 + f(k) where f(k) is something that goes to zero for large k. SymPy gives the same partial sum formula in terms of 2F1 (hyper) although it looks a little different with gamma functions: In [29]: print(summation(binomial(S(-2)/5, n), (n, 1, k))) (5/2 - 5*2**(3/5)/4)*gamma(3/5)/gamma(-2/5) - gamma(3/5)*hyper((1, k + 7/5), (k + 2,), -1)/(gamma(-k - 2/5)*gamma(k + 2)) I think SymPy and WolframAlpha are in agreement but just WA does not compute a closed form for this particular sum whereas SymPy does get the closed form but SymPy does not simplify the gamma functions as nicely as WA does. I think simplify here could be improved: In [33]: e = summation(binomial(S(-2)/5, n), (n, 1, oo)) In [34]: print(e) (5/2 - 5*2**(3/5)/4)*gamma(3/5)/gamma(-2/5) In [35]: e.evalf() Out[35]: -0.242141716744801 In [36]: 2**(-2/5)-1 Out[36]: -0.242141716744801 In [37]: print(simplify(e)) 5*(2 - 2**(3/5))*gamma(3/5)/(4*gamma(-2/5)) Maybe gammasimp could handle this better somehow. -- Oscar On Wed, 10 Sept 2025 at 12:19, Paul Royik <[email protected]> wrote: > > sum of binomial(-2/5, n), n=1..infinity > > SymPy says that the answer is 2^(-2/5)-1. > WolframAlpha says that the series is divergent. > > What answer is correct? > > -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To view this discussion visit > https://groups.google.com/d/msgid/sympy/16f407ca-97a8-4565-ace5-1157c2702722n%40googlegroups.com. -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion visit https://groups.google.com/d/msgid/sympy/CAHVvXxT7TFtihTaSroW7ftQSi1U_%3DhGEwA%3D1bNn1xieJjhTkqQ%40mail.gmail.com.
