Bug/problem evaluating infinite sum

[OHappyDay]

I tried to evaluate the infinite product:

prod((2^n-1)/2^n) (n=1,oo)

by converting the product to a sum via logarithm:

sum(log(1-2^-k),k,1,oo)

The sum (and thus the product) should, according to WolframAlpha, converge with a final value of about

-1.24206

This video (https://www.youtube.com/watch?v=KDyHJlNkov8) indicates that the product converges with an irrational value.

Sage reports that the sum is divergent.

Ideas? Is this again a failure in Maxima?

[David Joyner]

On Tue, Dec 10, 2024 at 8:43 AM 'OHappyDay' via sage-support <sage-s...@googlegroups.com> wrote:

I tried to evaluate the infinite product:

prod((2^n-1)/2^n) (n=1,oo)

by converting the product to a sum via logarithm:

sum(log(1-2^-k),k,1,oo)

The sum (and thus the product) should, according to WolframAlpha, converge with a final value of about

-1.24206

Sympy does it:

sage: from sympy import oo, Sum, log, Product

sage: p0 = Product((1-1/2^n), (n, 1, oo))

sage: p0.evalf()

0.288788095086602

sage: s0 = Sum( log(1-1/2^n), (n, 1, oo))

sage: s0.evalf()

-1.24206209481242

sage: exp(-1.242) ## check

0.288806027885956

 

This video (https://www.youtube.com/watch?v=KDyHJlNkov8) indicates that the product converges with an irrational value.

Sage reports that the sum is divergent.

Ideas? Is this again a failure in Maxima?

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[OHappyDay]

Thanks for investigating. I tried to use sympy this way:

    sum(log(1-2^-k),k,1,oo,algorithm="sympy")

but this failed for other reasons (NotImplementedError: conversion to SageMath is not implemented)

[Nils Bruin]

Investigating the error shows that sympy didn't do anything with this sum:

     671         try:
--> 672             return result._sage_()
    673         except AttributeError:
    674             raise AttributeError("Unable to convert SymPy result (={}) into"

ipdb> p result
Sum(log(1 - 1/2**k), (k, 1, oo))

so while solving the "NotImplemented" error would be nice, it wouldn't get you anything you didn't have already. The question then really is how to *numerically* evaluate an infinite sum in sage. As shown above, one way could be to use sympy for this.  It may be that numerical sum evaluation hasn't received much attention in sage. The link below also suggests using sympy:

https://ask.sagemath.org/question/32122/how-to-evaluate-the-infinite-sum-of-12n-1-over-all-positive-integers/

I'm not sure how rigorous these methods are. You may have to do some analysis yourself to guarantee an error bound.

[Emmanuel Charpentier]

A bit of exploration in the respective interpreters show that neither Maxima, Sage, Giac, Fricas, Sympy nor Mathematica can give an expression for this sum. Wolfram Alpha gives the numerical answer the OP reported, but I have been unable to get any explanation about the process used ; graphics appended to Wolfra Alpha's answer suggest that this numerical estimate is the numerical value of the sum for a "sufficiently large" value of k, with no indication about the choice of this "sufficiently large"...

As suggested by Nils Bruin, further work on numerical computation of sums is granted...

HTH,

[OHappyDay]

After contacting the Maxima team it turns out that Maxima indeed can evaluate the inifinite sum (at least a numerical value but not a closed form which probably does not exist)

Sage: 

k=var('k'); sum(log(1-2^-k),k,1,20) results in an error message ending with "Sum is divergent".

Maxima:

load ("levin"); 

bflevin_u_sum (log (1 - 1/2^k), k, 1);

This gives the numerical value  - 1.2420620948