prove an identity for any integer

asked 2021-02-13 21:40:36 +0200

rue82 gravatar image

Let $n$ be a positive integer and $m = (m_1, \ldots, m_n)$ an $n$-dimensional vector of real numbers. Let $g$ be a real number.

I want to prove, for any $n$ and $m$, an equality of the form $$ \sum_{i=1}^n f_i (m,g) = 0 $$ where the function $f_i$ is a rational function of $m$ and $g$.

Of course it's easy to check this by substituting finite values of $n$, but is there a way in Sage to prove it for any integer?

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Sage supports symbolic computations like in this example:

sage: k, m = var('k, m')
sage: sum(1/k^4, k, 1, oo)
sage: sum(binomial(m,k), k, 0, m)
Max Alekseyev gravatar imageMax Alekseyev ( 2021-02-13 22:01:08 +0200 )edit

Thanks, I did see that, although in my case it just prints the sums and does not simplify them to zero.

rue82 gravatar imagerue82 ( 2021-02-14 10:05:48 +0200 )edit

Then it's possible that your problem is out for reach for Sage and needs to be addressed by analytic rather than computational methods. You may ask for help at

Max Alekseyev gravatar imageMax Alekseyev ( 2021-02-14 16:40:39 +0200 )edit