# Question Video: Using the π-Series Test Mathematics • Higher Education

Use the π-series test to determine whether the series β_(π = 1)^(β) 1/4π is divergent or convergent.

01:51

### Video Transcript

Use the π-series test to determine whether the series the sum from π equals one to β of one divided by four π is divergent or convergent.

The question gives us an infinite series. It wants us to determine whether this series is divergent or convergent by using a π-series test. Letβs start by recalling what we mean by the π-series test.

We call the sum from π equals one to β of one divided by π to the πth power a π-series. And we know that this series is convergent when π is greater than one and divergent when π is less than or equal to one. A π-series test means to compare our series to a π-series. We can then determine the convergence or divergence by using this test.

So, we want to compare the series given to us in the question, thatβs the sum from π equals one to β of one divided by four π, with a π-series. To start, we can notice the π in our denominator can be rewritten as π to the first power. We can now see our series is almost in the form of a π-series. We just have this constant factor of four in our denominator. But this is a constant factor, so we can take this outside of our sum. This wonβt change the convergence or divergence of our series.

In other words, we can rewrite our series as one-quarter times the sum from π equals one to β of one divided by π to the first power. We can now see this is a constant multiple of a π-series where π is equal to one. And we know when π is equal to one, our π-series will be divergent. In fact, when π is equal to one, we call this series the harmonic series.

So, the series given to us in the question is one-quarter times our divergent series. This means the series given to us in the question is also divergent. Therefore, by using the π-series test, we were able to show the sum from π equals one to β of one divided by four π is divergent.