ALTEREDZINE: Limit of a sequence

Friday, July 1, 2022

Limit of a sequence

Introduction to the limit of a sequence

Let's consider the sequence:

We can have a graphic representation of this sequence. In order to do that we need to write the sequence as a function i.e f(n) = n + 1/n². We start by finding the first few points of the graph:

We plug the first values of n in the function in order to find the corresponding values of f(n)

For n = 1 we have f(1) = 1 + 1/(1)² = 2/1 = 2

For n = 2 we have f(2) = 2 + 1/(2)² = 3/4

For n = 3 we have f(3) = 3 + 1/(3)² = 4/9

For n = 4 we have f(4) = 4 + 1/(4)² = 5/16

For n = 5 we have f(5) = 5 +1/(5)² = 6/25

The first few points of the graph are: (1, 2), (2, 3/4), (3, 4/9), (4, 5/16), (5, 6/25).

The graph of the function looks like this for the first 30 points of the graph:

We observe that as the the value of n increases the value of f(n) decreases. For values of n sufficiently large, aⁿ becomes close to zero. We can express this observation like this:

When n→∞, aⁿ→0. This is nothing than the limit of aⁿ when n approaches infinity. When n approaches infinity the limit of aⁿ is 0. We can express this by the notation:

Definition of the limit of a sequence

If a sequence aⁿ becomes closer and closer to a number L when the value of n becomes sufficiently large, we say that the limit of aⁿ is L when n approaches infinity. We write this simply as:

Infinite limit of a sequence

If a sequence aⁿ becomes larger and larger whenever the value of n becomes sufficiently big, we say that the limit of aⁿ is positive infinity when n approaches infinity. We write:

If aⁿ becomes larger and larger negatively whenever the value of n becomes sufficiently big, we say that the limit of aⁿ is negative infinity when n approaches infinity. We write:

Precise definition of the limit of a sequence

1) We say that

if for every ε>0 there is an integer N such that

❘aⁿ -L❘＜ε whenever n>M

2) We say that

if for every number M>0 there is an integer N such that

aⁿ >M whenever n>N

3) We affirm that

if for every number M<0 there is an integer N such that

aⁿ <M whenever n>N

Convergence and Divergence of the limit of a sequence

If the limit of of a sequence when n approaches infinity is a finite number, the sequence is convergent. If the limit of a sequence when n approaches infinity is infinite positively or negatively, the sequence is divergent.

Properties of the limit of a sequence

The limit of a sequence is equivalent to the limit of a function since a sequence aⁿ can be written as f(n) = aⁿ. Therefore the properties of the limit of a function are also valid for the limits of a sequence

If aⁿ and bⁿ are convergent, we have the following properties: