Mathematically expressing the sum of odd and even numbers

Question

I am aware that if I want to write the product of a function $2N^2$ starting with numbers from $1$ to $n$, I write something like this $\prod_{N=1}^{n} 2N^2$, what if I want $N$ to start from 1 and step through odd numbers or even numbers? How do I express this? It is a simple question but I am getting somethings a bit mixed up.

Answer 1

If you want to step through the even numbers, you can write $\prod _{n=1}^k(2(2n)^2)$. If you want to step through the odd numbers, you can write $\prod _{n=0}^k(2(2n+1)^2)$. I don't know of a mathematical equivalent to the computer step as in for i=1 to 11 step 2 other than this.

I have seen the notation $$\sum_{\stackrel {p \le n}{p \text { prime}}}$$ for the sum of some expression over all the primes less than or equal to $n$.

Using the formula for even numbers makes it easier to deal with. In your example we can recognize the factorial and write $$\prod _{n=1}^k(2(2n)^2)=2^k(k!)^2\\ \prod _{n=0}^k(2(2n+1)^2)=2^{k+1}\left(\frac {(2n+1)!}{2^nn!}\right)^2$$