Helpful Tools for Calculus, Chapter 2: Helpful Tools for the Mathematics of Calculus
Previous Section: 2.4 Pascal’s Triangle and Binomial Expansion
Summation Notation
Finally, one last helpful tool for calculus is the idea of repeatedly adding things together. Summation is the idea of repeated addition. This could be as simple as adding one, over and over. It could be adding up fractions to see if there is a specific answer that results. We could even be counting shapes, functions, or operations being done repeatedly and added together. In calculus, summation plays a major role in identifying ways to problem solve and attack problems effectively.
Notation
Summation notation uses the capital sigma from Greek: 
. In mathematics, it has the following pieces,
The i=1 on the bottom can be thought of an initial value for the counter. Where does the summation begin? In this case, it begins with the 1st term, which is quite common. i is used as the index variable, so starting at 1 and so on. It is also common to begin summations with i=0 or to even use other variables like j or k for the index.
At the top of the sigma is the letter n. n represents the ending value for the index. This can be represented with a number, like 5, in order to say, ‘counting from 1 to 5’, or with an infinity to discuss an infinite sum. The letters m and n are common to use as a variable for this term.
f(i) represents the things being added, which is generally a function of the index. So, we add the first term, the second term, the third term, … etcetera.
As an example, we would write the sum of 5, 10 times, as
which means 5+5+5+5+5+5+5+5+5+5. Using a variable, the sum of the numbers from 1 to 10 is 1+2+3+4+5+6+7+8+9+10=
More generally,
Summation Rules
The mathematics of summation can be quite useful, especially as we discover shortcuts and formulas that help eliminate the tediousness of repeatedly adding or counting. One thing to take into account easily is the idea of factoring out a common number.
2+4+6+8+10=2(1+2+3+4+5)
Is easily written as
In general, this gives us what can be called the constant rule of summation:
Multiple Summations
It turns out that in many uses of summation, it is often helpful to add along multiple directions or with multiple variables. This leads to double or triple summations. These are often written like the following, we were vary the symbols to differentiate the variables:
Some things that are helpful include knowing that the order of summation does not matter and things can be factored out of a summation if it does not include the relevant variable.
