10.1. Sequences, Series, and Summation Notation
An important topic in mathematics is that of sequences. A sequence is an ordered set of numbers, each labeled with an index telling us where that number lies in the sequence. The index of a sequence usually starts at 0, though it can also start at 1 or other values. Thus, a standard sequence has the form a0, a1, a2,... . Sequences may be finite or infinite.
Example 1: For each of the following sequences, each beginning with a0, find a10 and a20.
- (a) The sequence of whole numbers: 0, 1, 2, 3, ...
- (b) The sequence of nonnegative even numbers: 0, 2, 4, 6, ...
- (c) The sequence of squares of whole numbers: 0, 1, 4, 9, ...
- (d) The sequence of odd numbers: 1, 3, 5, 7, ...
- (e) The sequence of primes: 2, 3, 5, 7, ...
Whenever possible, it is a good idea to write down a general formula for the kth term of any sequence. This way, one can compute an element of the sequence with arbitrary index.
- (a) The formula for the kth whole number is ak = k. Thus we have a10 = 10 and a20 = 20.
- (b) The formula for the kth nonnegative even number is ak = 2k. Thus we have a10 = 20 and a20 = 40.
- (c) The formula for the kth square is ak = k2. Thus we have a10 = 100 and a20 = 400.
- (d) The formula for the kth odd number is ak = 2k + 1. Thus we have a10 = 21 and a20 = 41.
- (e) There is no simple formula for the kth prime. The best way to compute it is to list all the primes up to that point. The first 21 primes are as follows: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, and 73. Thus, we see that a10 = 31, the 11th prime, and a20 = 73, the 21st prime.
For every sequence, there is an associated sequence known as a series. The series corresponding to a given sequence is the sequence of sums of consecutive terms of the sequence. Thus, for instance, the series associated with the sequence of whole numbers is 0, 0+1, 0+1+2, 0+1+2+3, ... = 0, 1, 3, 6, 10, ... . Finite sequences give rise to finite series and infinite sequences to infinite series.
When studying series, it is a good idea to have a compact way to represent a large sum of terms without having to write them all out. For this purpose, we use summation notation. A sum of terms of a sequence, i.e. a series, is indicated with the capital Greek letter sigma. Thus, we write the series associated with the sequence a0, a1, a2,... as Σak, which is shorthand for a0 + a1 + a2 + ... . If we only wish to add the terms with index up to n, we write
More generally, if we wish to add the terms with index ranging from m to n, we write
Example 2: Compute the following sums:
- (a) We have S = 1 + 2 + 3 + 4 + 5 = 15.
- (b) We have S = 6 + 7 + 8 + 9 + 10 = 40.
- (c) We have S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55.
- (d) We have S = 1 + 4 + 9 + 16 + 25 = 55.
- (e) We have S = 7 + 9 + 11 + 13 + 15 = 55.