Use the 𝑝-series test to determine whether the series the sum from 𝑛 equals one to ∞ of one divided by four 𝑛 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 𝑝-series test. Let’s start by recalling what we mean by the 𝑝-series test.
We call the sum from 𝑛 equals one to ∞ of one divided by 𝑛 to the 𝑝th power a 𝑝-series. And we know that this series is convergent when 𝑝 is greater than one and divergent when 𝑝 is less than or equal to one. A 𝑝-series test means to compare our series to a 𝑝-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, that’s the sum from 𝑛 equals one to ∞ of one divided by four 𝑛, with a 𝑝-series. To start, we can notice the 𝑛 in our denominator can be rewritten as 𝑛 to the first power. We can now see our series is almost in the form of a 𝑝-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 won’t change the convergence or divergence of our series.
In other words, we can rewrite our series as one-quarter times the sum from 𝑛 equals one to ∞ of one divided by 𝑛 to the first power. We can now see this is a constant multiple of a 𝑝-series where 𝑝 is equal to one. And we know when 𝑝 is equal to one, our 𝑝-series will be divergent. In fact, when 𝑝 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 𝑝-series test, we were able to show the sum from 𝑛 equals one to ∞ of one divided by four 𝑛 is divergent.