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1/2 of PI

I was wondering - how would I specify the interval (the amount that n increases each time) between terms? Is that possible? What if I want it to increase by, say, 2, each time instead of one.

In Python it would be n += 2 -> I'm looking for an equivalent...

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up vote 4 down vote accepted

You could do it two different ways. You could use a substack, like this

$$ \prod_{\substack{n=1 \\ 2 \mid n}}^\infty \frac{4n^2}{4n^2 -1} .$$

Or you could also let $n = 2m$ (or $n = 2m - 1$ if you want to start at $1$) and take the product as $m$ goes from $1$ to $\infty$, like this

$$ \prod_{m=1}^\infty \frac{4(2m)^2}{4(2m)^2-1} . $$

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There are some different ways mathematicians deal with this. For simple cases like this one, the most popular way is to write the product as

$$\prod_{n=1}^\infty \frac{4(2n)^2}{4(2n)^2 - 1}.$$

For a more general case, one can use the notation

$$\prod_{n \in S} \frac{4n^2}{4n^2 - 1},$$

where $S$ is a set that $n$ takes values from, for example $S = \{2n \mid n \in \mathbb{N}\}$. This way is a little bit problematic, since the order of the summation or multiplication must be understood from the context.

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