Limit Laws for Sequences

If  and are convergent sequences and c is a constant, then

EXAMPLE 10: Determine if the sequence  converges of diverges.

 Divide numerator and denominator by the highest power of n that occurs in the denominator and then use the Limit Laws.

EXAMPLE 11: Is the sequence  convergent or divergent?

  As in Example 10, we divide numerator and denominator by n :

because the numerator is constant and the denominator approaches 0. So  is divergent.

EXAMPLE 12: Calculate 

Notice that both numerator and denominator approach infinity. We can’t apply l’Hospital’s Rule directly because it applies not to sequences but to functions of a real variable. However, we can apply l’Hospital’s Rule to the related function   and obtain

EXAMPLE 13: Determine whether the sequence  is convergent or divergent.

If we write out the terms of the sequence, we obtain 

Since the terms oscillate between 1 and -1 infinitely often,  does not approach any number. Thus  does not exist; that is, the sequence   is divergent.

EXAMPLE 14: Evaluate   if it exists.

 SOLUTION We first calculate the limit of the absolute value:

The following theorem says that if we apply a continuous function to the terms of a convergent sequence, the result is also convergent.

EXAMPLE 15: Determine if the sequence   converges or diverges.

 does not exist      Diverges

EXAMPLE 16: Determine if the sequence   converges or diverges.

Therefore, we have  Converges.

The sequence    is convergent if   and divergent for all other values of .

EXAMPLE 17: Determine if the sequence   converges of diverges.

Converges