The Integral test for convergence and divergence of series

 The objectives of this post are:

1) Know what the integral test is about

2) Use it to determine if a series is convergent or divergent.

Integral test

We have studied so far different types of series: geometric, harmonic, telescopic. The study of these series show easily if they are convergent or divergent. At the beginning we state that the determination of the convergence or divergence of a series leads to the determination of the convergence or the divergence of the sequence of partial sums. This leads to determine the general term of the partial sums. This is something really complicated. Some other alternative methods make it easier to study the convergence or divergence of a series. The divergence test is one of them. Now we are going to use the Integral test which assimilates the series to an improper integral.

Theorem 

Let's suppose that f is a positive, continuous and decreasing function on the interval [k, n) and f(n) = aₙ then

Example 1. Assuming that all the conditions for the integral test are met determine if the following series converges or diverges.

Solution

Let's compare :

Let's calculate the integral:

The integral is convergent. Therefore the series is convergent

Example 2 . Same question with the series

Solution

In the example above we use the constant b as the upper limit of the integral. We can use any letter as the upper limit. In this example use the letter t.

Let's compare:

Let's calculate the integral on the second side:

We do the substitution u = lnx . Therefore du = (1/x)dx. Then:

Let's substitute the integral in the line above by its calculated value:

The integral diverges so is the given series.

Practice

Determine if the following series is convergent or divergent