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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 a

_{0}, a

_{1}, a

_{2},... . Sequences may be finite or infinite.

**Example 1: **For each of the following sequences, each beginning with a_{0}, find a_{10} and a_{20}.

- (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, ...

**Solution: **

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 a
_{k}= k. Thus we have a_{10}= 10 and a_{20}= 20. - (b) The formula for the kth nonnegative even number is a
_{k}= 2k. Thus we have a_{10}= 20 and a_{20}= 40. - (c) The formula for the kth square is a
_{k}= k^{2}. Thus we have a_{10}= 100 and a_{20}= 400. - (d) The formula for the kth odd number is a
_{k}= 2k + 1. Thus we have a_{10}= 21 and a_{20}= 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 a
_{10}= 31, the 11th prime, and a_{20}= 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 a

_{0}, a

_{1}, a

_{2},... as Σa

_{k}, which is shorthand for a

_{0}+ a

_{1}+ a

_{2}+ ... . 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)
- (b)
- (c)
- (d)
- (e)

**Solution: **

- (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.

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