A question on summation notation and pi notation for multiplication.

Question

As I am in high school, I know the basics to summation and pi notation. However when people put things other than numbers on the top and bottom of the summation, I do not understand what they mean. Can anyone elaborate on some symbols or things I should know.

Answer 1

That means that the index of summation has to satisfy some conditions.

For example, $\sum_{i=1}^n a_i$ can also be written $\sum_{1 \le i\le n} a_i $.

Another common idiom in summation is $f(n) =\sum_{d \mid n} g(d) $. This means that $f(n)$ is the sum of $g(d)$ for all $d$ that exactly divide $n$. This requires you to know that the expression "$d\mid n$" means that $d$ exactly divides $n$, so the sum is taken over all positive integers $d$ which exactly divide $n$.

More generally, the expression $f(n) =\sum_{p(n, d)} g(d) $ means the sum is taken over all $d$ such that $p(n, d)$ is true.

If you don not understand a particular notation (and, there are many), ask.

Answer 2

$$\sum_{i=1}^3f(i)=f(1)+f(2)+f(3)$$

So if we instead put $$\sum_{i=1}^af(i)=f(1)+f(2)+\cdots+f(a)$$ (this only makes sense if $a$ is a positive integer)