Use the 𝑝-series test to determine whether the series the sum for 𝑛 equals one to ∞ of seven 𝑛 cubed divided by five 𝑛 to the fourth power 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 the 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 power of 𝑝 a 𝑝-series. And we know this is convergent for all values of 𝑝 greater than one and divergent whenever 𝑝 is less than or equal to one. This means if we can compare our series to a 𝑝-series, we can use this to determine the convergence or divergence of our series.
So let’s see if we can directly compare the series given to us in the question with a 𝑝-series. First, we have the sum from 𝑛 equals one to ∞ of seven 𝑛 cubed divided by five 𝑛 to the fourth power. We can see our numerator and our denominator share a factor of 𝑛 cubed. So we’ll cancel this shared factor of 𝑛 cubed in our numerator and our denominator. This gives us the sum from 𝑛 equals one to ∞ of seven divided by five times 𝑛. We might not see how we can directly compare this with a 𝑝-series. However, remember that 𝑛 is actually equal to 𝑛 raised to the first power.
We can now see this is very similar to a 𝑝-series. We just have a factor of seven in our numerator and a factor of five in our denominator. But we can just write our summand as seven over five times one over 𝑛 to the first power. Then, we remember taking a constant factor outside of our series does not change whether our series is convergent or divergent. This gives us seven over five times the sum from 𝑛 equals one to ∞ of one divided by 𝑛 to the first power. This is seven-fifths times a series. In fact, this series is a 𝑝-series where 𝑝 is equal to one. And we know that a 𝑝-series when 𝑝 is equal to one must be divergent.
So what we’ve shown is the series given to us in the question is a constant multiple of a divergent series. And this means it must be divergent. Therefore, by using the 𝑝-series test, we were able to show the sum from 𝑛 equals one to ∞ of seven 𝑛 cubed divided by five 𝑛 to the fourth power is divergent.