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Demokurs: Approach to the Basics of Calculus - ENGLISH

Sequences - Basics

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This section contains the most important definitions about sequences. Through these definitions the general notion of sequences will be explained, but then restricted to real number sequences.


  
 

Definition 1 (Sequence). Let M be a non-empty set. A sequence is a function:

f : ℕ → M.

Occasionally we speak about a sequence in M .


Note: Characteristics of the set ℕ give certain characteristics to the sequence. Because ℕ is ordered, the terms of the sequence are ordered.


Definition 2 (Terms and Indices): A sequence is denoted ( a 1 , a 2 , a 3 , … ) or for short ( a n ) n instead of f ( n ) . The numbers a 1 ,a 2 ,a 3 , … ∈ M are called the terms of the sequence. Because of the mapping

f : ℕ → M n ↦→ an
we can assign a unique number n ∈ ℕ to each term. We write this number as a subscript and define it as the index; it follows that we can identify any term of the sequence by its index.
n 1 2 3 4 5 6 7 8 9 …

↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
a n a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 …

A few easy examples

Random sequence of coloured squares. At any one time only one term of the sequence is displayed.

Example 1: The sequence of natural numbers
The sequence ( a n ) n defined by a n := n, n ∈ ℕ is called the sequence of natural numbers. Its first few terms are:

a1 = 1, a2 = 2, a3 = 3,...

This special sequence has the property that every term is the same as its index.


Example 2: The sequence of triangular numbers
Triangular numbers get their name due to the following geometric visualization: Stacking coins to form a triangular shape gives the following diagram:


To the first coin in the first layer we add two coins in a second layer to form the second picture a 2 . In turn, adding three coins to a 2 forms a 3 . From a mathematical point of view, this sequence is the result of summing natural numbers. To calculate the 10th triangular number we need to add the first 10 natural numbers:


D10 = 1 + 2 + 3 + ...+ 9 + 10


In general form the sequence is defined as:


D n = 1 + 2 + 3 + … + ( n − 1) + n.


This motivates the following definition


Notation and Definition (sum sequence): Let ( a n ) n ,a n : ℕ → M be a sequence with terms a n , the sum is written:

∑n a1 + a2 + a3 + ...+ an−1 + an =: ak k=1

The sign ∑ is called sigma sign here the index k increases from 1 to n.

Sum sequences are sequences whose terms are formed by summation of previous terms.


Thus the n th triangular number can be written as:


n D = ∑ k n k=1

Example 3: Sequence of square numbers
The sequence of square numbers ( q n ) n is defined by: q n = n 2 . The terms of this sequence can also be illustrated by the addition of coins.

Interestingly, the sum of two consecutive triangular numbers is a square number. So, for example, we have: 3 + 1 = 4 and 6 + 3 = 9. In general this gives the relationship:


q = D + D n n n−1

Example 4: Sequence of cube numbers
Analogously to the sequence of square number, we give the definition of cube numbers as

3 an := n .

The first terms of the sequence are: (1 , 8 , 27 , 64 , 125 , … ).



 

If, as in Definition 1, M = ℝ, we obtain the following definition:

Definition 3 (Sequence of Numbers): A sequence of real numbers is a function

f : ℕ → ℝ.


When we speak about a sequence in this part of the course we mean in general a real sequence.


Note: In this course, we only consider sequences in ℕ , ℚ and ℝ.

If we delete arbitrarily many terms from a sequence but retain the order of the remaining sequence, then we obtain a so called subsequence. Mathematically this is defined as follows:


Various ways to represent sequences

 

Definition 4 (Subsequence): Given a sequence ( a n ) n and a strictly monotonically increasing

function

φ : ℕ → ℕ

we call the composition 2

(aφ(n))n

a subsequence of ( a n ) n .

H. Böhm: Relief 6 Punkte ... (1959)

Example of subsequences in art

Example 5: Let ( q n ) n with q n := n 2 be the sequence of square numbers

(1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100 … )

and define the function φ ( n ) = 2 n. The composition ( q 2 n ) n yields:
( q 2 n ) n = ( q 2 ,q 4 ,q 6 ,q 8 ,q 10 , … )


= (4 , 16 , 36 , 64 , 100 , … ) .

Definition 5 (Sequence of differences):
Given a sequence ( a n ) n = a 1 , a 2 , a 3 , … , a n , …; then

(an+1 − an)n := a2 − a1, a3 − a2, ...
is called the 1st difference sequence of ( a n ) n

The 1 st difference sequence of the 1 st difference sequence is called the 2 nd difference sequence. Analogously the n th difference sequence is defined.

 

Example 6: Given the sequence ( a n ) n with a n := n2+n-- 2, i.e.


( a n ) n = (1 , 3 , 6 , 10 , 15 , 21 , 28 , 36 , … )


Let ( b n ) n be its 1 st difference sequence. Then it follows that

( b n ) n = ( a 2 − a 1 ,a 3 − a 2 ,a 4 − a 3 , … )


= (2 , 3 , 4 , 5 , 6 , 7 , 8 , 9)


A term of ( b n ) n has the general form

bn = an+1 − an (n + 1)2 + (n + 1) n2 + n) = ------------------− -------- 22 2 2 = (n-+-1)-+-(n-+-1)-−-n--−-n- 2 (n2 + 2n + 1) + 1 − n2 = ---------------------- 2n + 2 2 = ------- 2 = n + 1.

New terms:



 

Short Revision Exercises
 

Exercise 1: Illustrate the sequence of square numbers in at least three different ways.


Exercise 2: Let ( π n ) n be the sequence of decimal places of the constant π. Is the finite sequence (1 , 4 , 1 , 5 , 9 , 2) a subsequence ( π n ) n ? Give a reason for your answer.