Summations and Sigma Notation


Carl Freidrich Gauss


Suppose we wish to find the following sum: $$1+2+3+4+\cdots+848$$ We might take a cue from a young Carl Freidrich Gauss, one of history's greatest mathematicians, who apocryphally added up the sum of the counting numbers from 1 to 100 when he was eight years old in his head -- possibly through the clever rearrangement and grouping of the numbers shown below $$(1+100) + (2+99) + (3+98) + \cdots + (50+51)$$ Each of the fifty pairs sums to 101, resulting in a total of $50 \times 101 = 5050$.

This technique can be easily generalized to yield a formula for the sum of the first $n$ positive integers. Notice that we can write this sum (which we denote by $S$ below) in two different ways: $$\begin{array}{ccccccccccccccc} S &= &1 &+ &2 &+ &3 &+ &\cdots &+ &n\\ S &= &n &+ &(n-1) &+ &(n-2) &+ &\cdots &+ &1\\ \end{array}$$ Then, upon adding these two equations together, we have $$2S = (n+1) + (n+1) + (n+1) + \cdots + (n+1)$$ Noticing that the right side is the sum of $n$ identical terms, we can write it as a product and then solve for $S$\\ $$2S = n(n+1)$$ $$S = \frac{n(n+1)}{2}$$

Clearly this formula makes short work of our earlier problem, right? $$1+2+3+4+\cdots+848 = \frac{(848)(849)}{2} = 359,976$$ The sum of the first 848 positive integers is indeed 359,976 -- but here's a strange question: "What if that wasn't what was meant by $1+2+3+4+\cdots+848$?"

There is a certain amount of ambiguity here, after all. We are relying on our ability to be suggestive enough with the first few terms of the sum to make the pattern clear, and the values of the rest of the terms obvious. However, what if some other pattern of numbers started out with the same first four numbers and ended with the same value of 848? Believe it or not, this can actually happen! Consider the sequence created by the following function $$f(n) = n^4-10n^3+35n^2-49n+24$$ $$\begin{array}{c|c} n & f(n)\\\hline 1 & 1\\ 2 & 2\\ 3 & 3\\ 4 & 4\\ 5 & 29\\ 6 & 126\\ 7 & 367\\ 8 & 848 \end{array}$$

Now this is obviously a contrived example, but relying on the reader to see the intended pattern behind the "dot, dot, dot" in such sums can cause problems -- especially when the pattern is complicated -- so mathematicians have developed a more precise way to describe these types of sums, called sigma notation.

For example, in sigma notation, we can write the sum of the first 5 squares of positive integers as $$\sum_{i=1}^{5} i^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2$$ The index variable, $i$, plays the role of a counter that starts at the value given below the greek letter $\Sigma$ (sigma), and ends at the value given above the sigma. We create the terms of the sum by plugging the values of our counter into the formula found to the right of the $\Sigma$, one after another.

Here's another example: $$\sum_{i=3}^{7} i(i+1) = (3 \cdot 4) + (4 \cdot 5) + (5 \cdot 6) + (6 \cdot 7) + (7 \cdot 8)$$ Isn't that much better than having to guess the pattern intended with the following? $$12 + 20 + 30 + \cdots + 56$$ Returning to our Gauss-inspired discovery regarding the sum of the first $n$ positive integers, we can now use this notation to describe what we found in a very clear and elegant way: $$\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$$