Infinite Sequences

A sequence of real numbers is a function whose domain is the set of positive integers and range a set of real numbers.

f : ℕ → ℝ

A sequence of complex numbers is defined similarly, with ℝ replaced by ℂ. A sequence can be thought of as a list of numbers written in a definite order:

a₀, a₁, a₂, a₃, ... , a_n, ...

The number a1 is called the first term, a₁ is the second term, and in general a_n is the nth term i.e. the real value f(n) associated to n. The sequence {a₀, a₁, ..., a_n} is generally indicated as {a_n}. It's important to remark the fact that a sequence (always infinite) may well have a finite range; for instance, 1, 0, 1, 0, ... is an infinite repeating sequence while its range is a set consisting of just two numbers: 0 and 1. Some example are

{a_n} = {(−1)ⁿ} = {−1, 1, −1, 1, −1, ...}<

{a_n} = {n2} = {0, 1, 4, 9, 16, ..}

{a_n} = {1/n} = {1, 1/2, 1/3, 1/4, ..}

{a_n} = {5} = {5, 5, 5} (constant sequence)

Its nth term is a_n = (−1)ⁿ and the range is {−1, 1}. Some sequences can be defined by giving a formula for the nth term.

a_n = n/(n + 1), {1/2, 2/3, 3/4, 4/5, ..., n/(n + 1)}

Definition 2.1.2. A sequence {a_n} is said

Bounded below if there exists a real number m such that a_n ≥ m, ∀n ∈ ℕ;
Bounded above if there exists a real number M such that a_n ≤ M, ∀n ∈ ℕ;
Bounded. A sequence which is bounded both above and below. Thus a sequence {a_n} is said to be bounded if there exist two real numbers m and M such that m ≤ a_n ≤ M ∀ n ∈ ℕ;
monotone increasing (or non-decreasing) iff a_n ≤ a_n+1 ∀n;
monotone decreasing (or non-increasing) iff a_n ≥ a_n+1 ∀n;
strictly increasing if a_n < a_n+1 ∀n;
strictly decreasing if a_n > a_n+1 ∀n;
monotone if it is either increasing or decreasing;
strictly monotonic if it is either strictly increasing or strictly decreasing;
alternating< if {a_n} changes sign alternately. (e.g. {(−1)ⁿ}. □

For example {(−1)ⁿ)} is bounded by the two numbers {−1, 1}; {1/n} is bounded: 1 is the upper limit and 0 is a lower bound; {n²} is bounded below; {(−2)ⁿ)} is not bounded.

The most important question about a sequence is whether it converges. We define this notion as follows.

Example 2.1.3 (AP). An arithmetic sequence or arithmetic progression (AP) is a sequence in which each term after the first term is formed by adding a constant number to the preceding term. Consider the sequence 1, 4, 7, 10,... where each term is obtained from the previous term by adding the constant value 3. The general form of a term in this sequence is a_n = 3n − 2. The general form of an arithmetic sequence is given by

a₁, a₁ + d, a₁ + 2d, a₁ + 3d, ..., a₁ + (n − 1)d

The nth term of an A.P. is given by

a_n = a₁ + (n − 1)d

The value a₁ is the initial term in the sequence, and the value d is the constant difference between a term and its successor. In the above example we have a = 1 and d = 3.

By mathematical induction method, we prove that hte sum S_n, of the first n terms of arithmetic progression is calculated by the formula

S_{n} = \frac{a_{1} + a_{n}}{2} n

For n = 1, we obtain S₁ = (1/2) (a₁ + a₁) = a₁.
Assume that the formula for S_n holds for all positive integers n ≥ 1, then we'll prove the formula for n + 1, that is
$S_{n + 1} = \frac{a_{1} + a_{n + 1}}{2} n + 1$
We have that S_n+1 = S_n + a_n+1 and according to the inductive hyphothesys S_n =(a₁ + a_n)n/2. Then
$\begin{aligned} S_{n + 1} & = \frac{a_{1} + a_{n}}{2} n + a_{n + 1} = \frac{1}{2} [(a_{1} + a_{n}) n + 2 a_{n + 1}] = \\ = \frac{1}{2} [(a_{1} + a_{1} + (n - 1) d) n + 2 (a_{1} + n d)] = \frac{1}{2} [2 a_{1} (n + 1) + n (n + 1) d] = \\ = \frac{1}{2} (a_{1} + a_{1} + n d) (n + 1) = \frac{a_{1} + a_{n + 1}}{2} (n + 1) . \end{aligned}$
completing the proof. ■

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