Sum

The sum of two numbers is what we get when we add the two numbers. There are a number of ways of writing sums, with the most common being:

• Addition ($2+4+6 = 12$)
• Summation ($\sum_{k=1}^3 k = 1+2+3=6$)
• Computerization:

Sum = 0

For I = M to N

Sum = Sum + X(I)

Next I (in Visual BASIC)

Sigma notation

Sigma notation is a mathematical notation to write long sums in a short way. Sigma notation uses the Greek letter Sigma, $\sum$, and takes upper and lower bounds which tell us where the sum begins and where it ends. The lower bound usually has a variable (called the index) given a value, such as "i=2". This tells us that the summation begins at 2, and goes up by 1 until it reaches the number on the top.

Properties

$\sum_{i=1}^n 0 = 0$

$\sum_{i=1}^n 1 = n$

$\sum_{i=1}^n n = n^2$

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

$\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$

$\sum_{i=1}^n i^3 = \frac{n^2 (n+1)^2}{4}$

Applications

Sums are used to represent series and sequences. For example,

$\sum_{i=1}^4 \frac{1}{2^i} = \frac{1}{2^1} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4}$

The geometric series of a repeating decimal can be represented in summation,

$\sum_{i=1}^\infty \frac{3}{10^i} = 0.333333... = \frac{1}{3}$

The concept of an integral is a limit of sums. The area under a shape being defined as:

$\lim_{n \to \infty} \sum_{i=1}^n f(x_i^*)\Delta x$