Mathematics program                                            Sequences and series

  • Introduction

A sequence is a set of real numbers. It is a function, which is defined for the positive integers. The value of the function at a given integer is a term of the sequence. The range of a sequence is the collection of terms that make up the sequence. If the sequence only be defined for the positive integers up to a given integer n is called infinite sequences. If a sequence defined only for positive integers up to a certain integer are called finite sequences. A common way to represent a sequence is to lie out the members of a sequence in a list with a first member second member and so on as the example shown below:

0,  1,  2,  3,  4,  5…..

In this sequence the first term is 0, the second term is 1, and so on.

Notice that sequence doesn’t have to be defined by a sensible formula. Let’s see the sequence shown below:

3,  3.1,  3.14,  3.141,  3.1415,  3.14159,  3.141592…

As we observe that the sequence shows the value of π more and more accurately. It neither has a general term or a formula.

 

Two important categories of sequences are  sequences, and . Both are examples of a , a sequence in which each term, besides of the first term, depends on the previous term. Both of these types of sequences will be discussed.

When the terms of a sequence are summed, the result is called a . Some series increase without bound as n increases, but others approach a limit. Those that do approach a limit are called convergent series, and those that don't are called divergent series. Divergent series sometimes increase without bound, sometimes decrease without bound, and other times oscillate between values without approaching a single limit. There are also certain formulas for calculating the limits of series that I will focus on. The study of series is an important part of calculus, and it all starts with sequences.

  • Sequences including those given by a general formula for the nth term

As I have just mentioned pervious, the nth term of a sequence can be used to find any term in the sequence. When the general term of a sequence is given in terms of n we can find the first, second, third and fourth terms of the sequence by substituting n = 1, n = 2, n = 3, and n = 4 in the general term.

Example 1

Find the first 3 terms and the 10th term of a sequence has the general term 2n + n.

Take n = 1, 2, 3, and 4 in turn to find the terms

a1 = 2 × 1 + 1 = 3

a2 = 2 × 2 + 2 = 6

a3 = 2 × 3 + 3 = 9

Take n = 10 to find the 10th term

a10 = 2 × 10 + 10 = 30

Thus, 3, 6, 9 are the first three terms of the sequence and 30 is the 10th term of the sequence.

Example 2

Known that a1 = 1, an+1 = 2an / (an + 2) (n ∈ N), write the first 5 terms of the sequence and summary the general term.

a1 = 1, a2 = 2/3, a3 = 1/2, a4 = 2/5, a5 = 1/3.

Observe each terms, we can unify the numerator as 2:

a11 = 2/2

a2 = 1/2 = 2/4

a3 = 1/3 = 2/6

Suppose that the general term is an = 2 / (n + 1)

So the first 5 terms of the sequence are 1, 2/3, 1/2, 2/5, 1/3; the general term of the sequence is 2 / (n + 1).

Example 3

Given that an = 9n (n + 1) / 10n (n ∈ N), is there a biggest term in the series {a}? If it has, find the biggest term.

Join now!

Suppose that the nth term is the biggest term is n in {a}, thus

an ≤ an-1  

an ≤ an+1

Namely

9n (n + 1) / 10n ≤ 9n-1 n / 10n-1

9n (n + 1) / 10n ≤ 9(n + 1)(n + 2) / 10n+1

So, 8 ≤ n ≤ 9 which means a8 and a9 are the biggest terms.

  • From the first several terms we can summary a general term of the sequence. But it may not just have one probability.

Example 4

The first 4 terms are 1, 2, 4, ...

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