If you replace -2/5 with -0.4, WolframAlpha says that the series diverges: https://www.wolframalpha.com/input?i=sum%28binomial%28-0.4%2C+n%29%2C+%28n%2C+1%2C+inf%29%29
On Wednesday, September 10, 2025 at 2:48:04 PM UTC+3 Oscar wrote: > 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 <distan...@gmail.com> 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 sympy+un...@googlegroups.com. > > 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 sympy+unsubscr...@googlegroups.com. To view this discussion visit https://groups.google.com/d/msgid/sympy/b73c1f9f-774f-451f-9574-b0db731cae7dn%40googlegroups.com.
