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.

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