I think WolframAlpha is wrong when it says that it diverges. My guess
is that since you use a float it uses a heuristic numeric check for
convergence and since this converges slowly the heuristic check looks
like it does not converge. You can see mpmath struggles with it as
well:
In [1]: f = binomial(-S(2)/5, 1 + k) * hyper([1, 7/5 + k], [2 + k], -1)
In [2]: f.evalf(subs={k:1e10})
...
File <string>:38, in hypsum_2_1_RZ_Z_R(coeffs, z, prec, wp, epsshift,
magnitude_check, **kwargs)
NoConvergence: Hypergeometric series converges too slowly. Try
increasing maxterms.
On Wed, 10 Sept 2025 at 12:50, Paul Royik <[email protected]> wrote:
>
> 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 <[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?
>> >
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