Sigma Notation (Summation).

Question

I have recently started out with series and I have a few set of questions.

Consider, $$\sum\limits_{k=p}^{q} f(k) $$

1. How is the notation read as? Like summation $k = p $ to $q$ or Sigma $k=p$ to $q$.

2. Why (I want to know the reason pls) should $k$ always be an integral value?

3. Can $k$ take negative values like $-3$, $-7$? or fractional values like $ \frac{1}{2} $ etc?

I tried googling but all I found was all sorts of irrelevant or advanced stuffs. Please use laymen terms :)

Answer 1

1. makes no difference, pick whatever is more convenient
2. we usually think of $\sum$ as a discrete operator, letting $k$ be real, for example, will transform it into a continuous one, alike the $\int$.
3. As is clear from (2), $k$ can take values in any discrete set, but the convention is to have it vary over subsets of integers. So you may encounter $$ \sum_{x \in A} f(x). $$

Answer 2

How is the notation read as? Like summation k=p to q or Sigma k=p to q.

Sigma f(k), from p to q.

Why (I want to know the reason pls) should k always be an integral value?

This is by the definition of $\sum $ notation for adding countable quantities.

Can $k$ take negative values like −3, −7? or fractional values like 1/2 etc?

Yes, you cans start with negative integers as well, but not fractions.