Silhouette wrote:Magnus Anderson wrote:Silhouette wrote:Every partial product is greater than zero, yes - because it's only partial! Thus the same logic carried over to any "non-partial-product" of some kind of "infinite product" is invalid. Again - poor logic skills on your part.

Every partial product of \(0 \times 0 \times 0 \times \cdots\) is equal to \(0\) and that's precisely why its result is equal to \(0\).

The same does not apply to \(\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \cdots\).

I can repeat myself too if you want.

That's not getting us anywhere though is it.

I know every partial product is greater than 0. This just panders to your appeal to finitude that you keep denying you're doing.

INFINITES. We're dealing with them, not "partial, finitely somewhere along the way of an infinite series" which you're pretending still counts as valid for evaluating an infinite series.

\(0 \times 0 \times 0 \times \cdots\) is an infinite product. It's a product made out of an infinite number of terms. It has no end.

Agree?

Even though \(0 \times 0 \times 0 \times \cdots\) is a product made out of an infinite number of terms, its result is equal to a finite number that is \(0\).

Agree?

You are certainly not telling us that the result of an infinite product can never be calculated because due to the number of terms being endless there is no end to the process of calculation? I hope that's not what you're trying to tell us.

How do we know that the result of this infinite product is \(0\)? We know it's \(0\) because we know that all of its partial products are equal to \(0\). What this means is that we can calculate the result of an infinite product by looking at its partial products.

So yes, I know full well we're dealing with an infinite product (and not merely its partial products.) But I also know that the way to calculate the result of an infinite product (or at least, to calculate its bounds) is by analyzing its partial products.

The insight is pretty basic: regardless of how many terms there are in a product of the form \(\Pi^{n} 0\), the result will always be \(0\). Now matter how big the number of terms is (indeed, even if its bigger than every finite number), the result will be \(0\).

Something similar applies to \(\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \cdots\). Here, we can observe that every partial product has a different value. It doesn't matter how large the number of terms is (even if its larger than every finite number), the result is always different from every other partial product. This alone tells us that the result is not equal to \(0\). But there is one more thing we can observe: no matter how large the number of terms is, the result is never equal to or below \(0\). And this is what tells us that the result is greater than \(0\).

You, on the other hand, are telling us that we can get \(0\) by taking \(\frac{1}{10}\) and raising it to a sufficiently large number (namely, that of infinity.)

The point is that any product of the form \((\frac{1}{10})^n\) where \(n\) is any number greater than \(0\) is greater than \(0\). The only condition for \(n\) is to be greater than \(0\). It can be a finite number but it can also be a number larger than every finite number (i.e. an infinite number.) Either way, the result is greater than \(0\).