## What Is A Summation?

The mathematical expression

$\sum_{k=1}^{100} k$

means

$1 + 2 + 3 + \ldots + 100.$

More generally,

$\sum_{k=m}^n f(k)$

means

$f(m) + f(m+1) + f(m+2) + \ldots + f(n)$.

But what if you don’t understand what “+ … +” means? Sure, most people understand what that means intuitively, but how could you define “…”?

From a logical point of view, you could instead make a recursive definition

$\sum_{k=m}^m f(k) \equiv f(m)\, \, , \quad m \in \mathbf{Z}$

and

$\sum_{k=m}^n f(k) \equiv f(n) + \sum_{k=m}^{n-1}f(k)\,\, , \quad (n>m) \land (m,n \in \mathbf{Z})$.

Most of the time, when I want to prove a summation, I prove it by induction anyway, so this definition is actually helpful. Suppose we want to prove

$\sum_{k=1}^n k = \frac{n(n+1)}{2}.$

For $n=1$,

$\sum_{k=1}^1 k = 1$ by definition.

Supposing the summation to work for $n-1$, for the case $n$ we have

$\sum_{k=1}^n k = n + \sum_{k=1}^{n-1}k = n + \frac{(n-1)n}{2} = \frac{n(n+1)}{2}$

where the first equation comes from the recursive definition of the summation, the second by hypothesis, and the third by algebra.