I hope that Product Convergence will be useful, also. So i will be making a PR about the

`.is_convergent()`

in `Product`

in SymPy. This will be quite simple to make, but since i will also be adding some latex in docs(which is quite new to me). So will that be useful to have `is_convergence`

for Productss as well.?
I have been thinking of the connection between products $\prod (1 + a_n)$ and sums $\sum a_n$. If all the terms $a_n$ are positive or negative, then the product and the sum both converge or both diverge. (For negative $a_n$, the product is considered divergent if is tends to 0.) However, in general it is possible that the sum may converge even if the product is divergent.

Consider, for example, the convergent alternating series $\sum (-1)^n/\sqrt{n}$. Grouping the terms of the product pairwise we get

$(1 - 1/\sqrt{2n - 1})(1 + 1/\sqrt{2n}) < (1 - 1/\sqrt{2n})(1 + 1/\sqrt{2n}) = 1 - 1/2n$

showing that the product is divergent.

Hence, it would probably be best to consider the absolute convergence of the sum. It will give a sufficient but not necessary condition for the convergence of the product.

$(1 - 1/\sqrt{2n - 1})(1 + 1/\sqrt{2n}) < (1 - 1/\sqrt{2n})(1 + 1/\sqrt{2n}) = 1 - 1/2n$

showing that the product is divergent.

Hence, it would probably be best to consider the absolute convergence of the sum. It will give a sufficient but not necessary condition for the convergence of the product.

Sorry, to interrupt, i have not read your two mentioned points. $\sum_{n=2}^{\infty} \log(\frac{1}{n})$ (sum diverges) while $\prod_{n=2}^{\infty} \frac{1}{n}$ (produc converges)

So i guess i should use

`solveset`

to check for values of `abs(sequence)`

are all $< 1$ then return True.
The product $\prod 1/n$ is usually considered divergent.

It is usually assumed that the terms are nonzero.

To be clear for definition of wikipedia: "The product of positive real numbers $\prod_{n=1}^{\infty} a_{n}$ converges to a non-zero real number if and only if the sum $\sum_{n=1}^{\infty} \log{a_{n}}$ converges."

If any one of the terms is zero, all partial products are zero. That is not accepted as convergent. Moreover, the product should be convergent if and only if the product of the inverses is convergent. (So $\prod n$ is divergent.)

Yes, non-zero is the point.

Perhaps we could 'generalize' the condition of convergence in such a way that a finite number of the terms could be zero. Then only the tail product would decide the convergence.

Hence, it might not be necessary to test

`abs(term) < 1`

.
Yes, thanks for the explanation. It seems more clear now. I will do this point then.

This article also gives justification of your point.

Can you please give an example of this? I will be more clear.