Some theorems that facilitate the calculations of the limit of a sequence

 Here are some theorems that facilitate the calculations of the limit of a sequence: 

Squeeze theorem

The expression "for all n>N for some N" means that the theorem is valid for all sufficiently large n but if some values aren't large for the first few values of n, the theorem is still valid.

Theorem 1

It's not possible to take the limits of all sequences that can be written as functions. This is especially true for  sequences that alternate in sign. The following theorem will help in this regard:

Theorem 2

The sequence {rₙ} varying from 0 to +∞ converges if -1<r≤1 and diverges for all other values of r.

= 0 if  -1<r<1 or 1 if r = 1

Exercises:

Find the limit of the following sequences:

a)  

b)  

Solutions

a) Let's calculate 

b) According to theorem 2 a sequence of this type converges if -1≤r≤1 and diverges for all other values. Here r = -1 the sequence converges.

Practice

Find the limit of the sequence {2ⁿ} varying from 0 to +∞

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:

Exercises. Determine if the following sequences converge or diverge. If the sequence converges, determine its limit.

  

Solutions

Practice

Determine if the following sequences converge or diverge