Considerations on convergence and divergence of series

 In a previous post, to determine if a sequence is convergent or divergent we had to find the limit of the sequence of partial sums  . To do this, we had to determine the general term of the sequence of partial sums and find its limit. This is a tedious work to find the formula of the general term of the partial  sums. There are different other ways to determine if a series is convergent or divergent. One of them is called the Divergence Test. We will explore the other ways later. Before exploring the divergence test, let's see what happens when a series converges. 

Theorem. If Σaₙ converges then lim aₙ = 0 when n approaches infinity. 

This statement means that if a series converge, then lim aₙ will necessarily be equal to zero when n approaches infinity. The converse of this statement is not true. This means that if  lim aₙ = 0 when n approaches infinity, the series doesn't necessarily converge. There are series of which the limit is equal to zero but they are not necessarily convergent. 

 Divergence Test.

The divergence test allows to quickly conclude if a series is divergent. But if the limit is equal to zero, we don't know if the series is convergent or divergent. We say that the divergence test is inconclusive. The divergence test is stated like this: If lim a #0 when n approaches infinity then Σ aₙ diverges

Example. Determine if the following series is convergent or divergent.

Let's calculate lim aₙ when n approaches infinity.

The limit is different of zero, therefore the series diverges.

Practice