## 24 Nov 2008

### Summation Notation and Simple Rules of Summation in Edwards & Penney

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presentation of Edwards & Penney's work, by Corry Shores

Edwards & Penney's Calculus is an incredibly-impressive, comprehensive, and understandable book. I highly recommend it.

Summation notation provides a concise representation for the sums of numerical sequences, and it takes the form:

The Σ is the Greek letter sigma, which means we are summing a sequence of terms. This whole sequence is signified by the ai. The i (called the summation variable, summation index, or running index) is the variable part of the terms, and it is substituted firstly with a 1 (which is what the i = 1 means), and it is subsequently substituted with the successive integers. The sequence ends when the i value reaches the n value.

For example:

Here we see that the summation variable is substituted firstly with a 1, and each substitution is squared. The substitutions continue until reaching 10. Then all the terms are added, to produce 385. The variable-letters are arbitrary, so we may note it different ways:

The notation might be labeled thus:

Simple Rules of Summation:

1)

2)

The sum of the kth powers of the first n positive integers

occurs often in area calculations. The values of this sum for k = 1, 2, and 3 are given in the following formulas:

k = 1:

k = 2:

k = 3:

Example: The sum of the first 10 positive integers is given by this equation

with n = 10:

The sum of this series' squares is given by the equation

and hence is

The sum of their cubes is given by the equation

And thus is

from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, p.290b-291d.