# Incorrect result for `sum(1/factorial(n**2),n,1,oo)`

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### Georgi Guninski

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Feb 12, 2024, 9:53:38 AMFeb 12
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There is discussion about this on mathoverlow [1]:

The closed form of `sum(1/factorial(n**2),n,1,oo)` doesn't appear
correct and it contradicts numerical computations, including verification
with mpmath.

Session:

sage: import mpmath
sage: su4=sum(1/factorial(n**2),n,1,oo);su4
hypergeometric((1,), (-I + 1, I + 1, -I*sqrt(2) + 1, I*sqrt(2) + 1), 1)
sage: CC(su4)
1.17227289255719 - 7.88860905221012e-31*I
sage: mpmath.hyper((1,), (-I + 1, I + 1, -I*sqrt(2) + 1, I*sqrt(2) + 1), 1)
mpc(real='1.1722728925571919', imag='-6.9025329206838533e-31')
sage: su5=sum(1/factorial(i**2) for i in range(1,100))
sage: CC(su5)
1.04166942239864

sage: mpmath.nsum(lambda n: 1/mpmath.gamma(1+n**2),[1,mpmath.inf])
mpf('1.0416694223986369')

[1]: https://mathoverflow.net/questions/463964/factorial-series-jd-sum-n-1-infty-frac1nd-and-hypergeometric-fu

### Dima Pasechnik

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Feb 14, 2024, 12:52:30 PMFeb 14
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It appears to come from Maxima, but I have trouble reproducing this in Maxima.
Perhaps it's a bug in the Maxima interface?
Is there a direct way to see how Maxima is called in this instance?

Dima

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### Oscar Benjamin

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Feb 14, 2024, 1:12:53 PMFeb 14
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Maxima's simplify_sum function produces something similar looking:

(%i4) load("simplify_sum");
(%o4) "/usr/share/maxima/5.45.1/share/solve_rec/simplify_sum.mac"

(%i5) sum(1/factorial(n^2), n, 1, inf), simpsum;
(%o5) 'sum(1/(n^2)!,n,1,inf)

(%i6) simplify_sum(%);
1/'product(n^2+%,%,1,2*n+1) non-rational term ratio to nusum
1/'product(n^2+%,%,1,2*n+1) non-rational term ratio to nusum
(%o6) %f[1,4]([1],[1-%i,%i+1,1-sqrt(2)*%i,sqrt(2)*%i+1],1)

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Oscar
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### Dima Pasechnik

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Feb 14, 2024, 2:14:36 PMFeb 14
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On Wed, Feb 14, 2024 at 6:12 PM Oscar Benjamin <oscar.j....@gmail.com> wrote:
Maxima's simplify_sum function produces something similar looking:

(%i4) load("simplify_sum");
(%o4) "/usr/share/maxima/5.45.1/share/solve_rec/simplify_sum.mac"

(%i5) sum(1/factorial(n^2), n, 1, inf), simpsum;
(%o5) 'sum(1/(n^2)!,n,1,inf)

(%i6) simplify_sum(%);

Oh, I see - I missed an explicit call to simplify_sum.

1/'product(n^2+%,%,1,2*n+1) non-rational term ratio to nusum
1/'product(n^2+%,%,1,2*n+1) non-rational term ratio to nusum
(%o6) %f[1,4]([1],[1-%i,%i+1,1-sqrt(2)*%i,sqrt(2)*%i+1],1)

It seems to be a bug in  simplify_sum() - nusum can't do it ( "non-rational term ratio to nusum")
indeed, it's obvious that the sum is not hypergeometric, as the consequent terms ratio is not of the right type,
so it does try something more clever - but fails.

Dima

### Dima Pasechnik

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Feb 14, 2024, 7:27:47 PMFeb 14
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### Georgi Guninski

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Apr 26, 2024, 9:24:27 AMApr 26
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On Thu, Feb 15, 2024 at 2:27 AM Dima Pasechnik <dim...@gmail.com> wrote:
>
> I've filed https://sourceforge.net/p/maxima/bugs/4262/
>

Is maxima supported?
There is no progress on their bug system for more than 2 months.
SEGV is not pleasant, but incorrect symbolic result casts doubts about
all symbolic sage computations, especially those that can't be
verified numerically.

### Dima Pasechnik

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Apr 26, 2024, 2:29:42 PMApr 26
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On Fri, Apr 26, 2024 at 2:24 PM Georgi Guninski <ggun...@gmail.com> wrote:
>
> On Thu, Feb 15, 2024 at 2:27 AM Dima Pasechnik <dim...@gmail.com> wrote:
> >
> > I've filed https://sourceforge.net/p/maxima/bugs/4262/
> >
>
> Is maxima supported?
> There is no progress on their bug system for more than 2 months.

not many people are involved, and some bugs stay open for years there.

> SEGV is not pleasant, but incorrect symbolic result casts doubts about
> all symbolic sage computations, especially those that can't be
> verified numerically.
>
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