Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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...

share|improve this question

2 Answers 2

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

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

share|improve this answer

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.