Simple Question about mathematical convention For expressions in the form: $$\sum_{i=1}^{k}f(i),$$ does this preclude the possibility that $k$ can be non positive as well as non-integer? Or more explicitly, can I make an induction on $k$? and take $1$ as base case? And then show that it is true for $k$ if it is true for $k-1$ ? Thank You.

This format does preclude non-integer numbers. It specifies that $i$ takes the integers between $0$ and $k$ inclusive, so yes, you could induct on $k$.

If we want to take non integer values, we can do that by specifying an "indexing set" $I$ and saying that $I$ contains all the values we want. We then write $\displaystyle \sum_{i \in I}f(i)$ to say that $i$ takes all values in the set $I$. This a commonly used in, for example, probability, where we take $I$ to be the set of all possible events and $f(i)$ the probability that $i$ occurs.