# Tagged Questions

For questions about the evaluation of finite products, or their properties. For infinite ones, use "infinite-products" tag.

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### how to estimate $\prod_{k=2}^n \log(k)$?

I wonder if I can estimate $\prod_{k=2}^n \log(k)$ as $a^l$ for some a. I know that it is bounded by $e^{n^2}$, but I would like to get something finer.
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### Proving $\prod_{i=2}^{i=n} \left(1-\frac{1}{i^2}\right) = \frac{n+1}{2n}$ by induction

So I have to prove the following using induction. ${\displaystyle \prod_{i=2}^{i=n} \left(1-\frac{1}{i^2}\right)} = \frac{n+1}{2n}$ I showed the basis step that if $n=i=2$, then the two functions ...
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### Find the product of $(1+\frac{1}{5^{2^n}})$ for $n=0$ to $6$

i.e. $(1+\frac{1}{5})(1+\frac{1}{25})(1+\frac{1}{625})...(1+\frac{1}{5^{64}})$ I have no idea or clue to this problem. Is there a general trick/procedure to this kinds of problem?
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### Is this a finite product describing the partial harmonic series sums?

http://mathworld.wolfram.com/EulerProduct.html In the second last line, it gives a product P(n). Is this supposed to be describing the finite terms of the harmonic series sum? I don't see how it does....
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### Good example showing why limits must exist in limit product rule

I'm looking for a way to show my calc 1 students not to use the limit laws without knowing that the individual limits exists. I could use $$\lim_{x\to 0} x^{2} \sin(1/x),$$ but by doing it wrong, one ...
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### For $n \in \mathbb{N}$, find and prove a formula for $\sum_{i=1}^n \frac{1}{i(i+1)}$, plus related question.

I was fairly easily able to obtain and prove this formula for the sum: $$S(n)=\frac{n}{n+1}$$ by typical means of computing the partial sums, observing the pattern, and proving by induction. My ...
### $(C \times D) \cap (A \times B) \neq \emptyset \implies C \cap A \neq \emptyset$ and $D \cap B \neq \emptyset$
If $C$ and $D$ are open sets, and $(C \times D) \cap (A \times B) \neq \emptyset$, then why is it necessarily true that $C \cap A \neq \emptyset$ and $D \cap B \neq \emptyset$?