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 +∞