Sequences and Series

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Sequences and Series

A sequence is an ordered listing of numbers such as {1, 3, 5, 7, 9}. In mathematical terms, a sequence is a function whose domain is the natural number set {1, 2, 3, 4, 5, …}, or some subset of the natural numbers, and whose range is the set of real numbers or some subset thereof. Let f denote the sequence {1, 3, 5, 7, 9}. Then f (1) = 1, f (2) = 3, f (3) = 5, f (4) = 7, and f (5) = 9. Here the domain of f is {1, 2, 3, 4, 5} and the range is {1, 3, 5, 7, 9}.

The terms of a sequence are often designated by subscripted variables. So for the sequence {1, 3, 5, 7, 9}, one could write a ₁ = 1, a ₂ = 3, a ₃ = 5, a ₄ = 7, and a ₅ = 9. A shorthand designation for a sequence written in this way is {a _n}. It is sometimes possible to get an explicit formula for the general term of a sequence. For example, the general nth term of the sequence {1, 3, 5, 7, 9} may be written a _n = 2n − 1, where n takes on values from the set {1, 2, 3, 4, 5}.

The sequence {1, 3, 5, 7, 9} is finite, since it contains only five terms, but in mathematics, much work is devoted to the study of infinite sequences.* For instance, the sequence of "all" odd natural numbers is written {1, 3, 5, 7, 9, …}, where the three dots indicate that there is an infinite number of terms following 9. Note that the general term of this sequence is also a _n = 2n − 1, but now n can take on any natural number value, however large. Other examples of infinite sequences include the even natural numbers, all multiples of 3, the digits of pi (π), and the set of all prime numbers.

*A famous infinite sequence is the so-called Fibonacci sequence {1, 1, 2, 3, 5, 8, 13, 21, …} in which the first two terms are each 1 and every term after the second is the sum of the two terms immediately preceding it.

The amount of money in a bank account paying compound interest at regular intervals is a sequence. If $100 is deposited at an annual interest rate of 5 percent compounded annually, then the amounts of money in the account at the end of each year form a sequence whose general term can be computed by the formula a _n = 100(1.05)ⁿ, where n is the number of years since the money was deposited.

Series

A series is just the sum of the terms of a sequence. Thus {1, 3, 5, 7, 9} is a sequence, but 1 + 3 + 5 + 7 + 9 is a series. So long as the series is finite the sum may be found by adding all the terms, so 1 + 3 + 5 + 7 + 9 = 25. If the series is infinite, then it is not possible to add all the terms by the ordinary addition algorithm , since one could never complete the task.

Nevertheless, there are infinite series that have finite sums. An example of one such series is: …, where, again, the three dots indicate that this series continues according to this pattern without ever ending. Now, obviously someone cannot sit down and complete the term by term addition of this series, even in a lifetime, but mathematicians reason in the following way. The sum of the first term is 1; the sum of the first two terms is 1.5; the sum of the first three terms is 1.75; the sum of the first four terms is 1.875; the sum of the first five terms is 1.9375; and so on. These are called the first five partial sums . Mathematically, it is possible to argue that no matter how many partial sums one takes, the n th partial sum will always be slightly less than 2 (in the above example), for any value of n. On the other hand, each new partial sum will be closer to 2 than the one before it. Therefore, one would say that the limiting value of the sequence of partial sums is 2 and the sum is defined as the infinite series …. This infinite series is saidto converge to 2. A case where the partial sums grow without bound is described as a series that diverges or that has no finite sum.