### Video Transcript

Use the π-series test to determine whether the series the sum for π equals one to β of seven π cubed divided by five π 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 π-series test. Letβs 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 power of π a π-series. And we know this is convergent for all values of π greater than one and divergent whenever π is less than or equal to one. This means if we can compare our series to a π-series, we can use this to determine the convergence or divergence of our series.

So letβs see if we can directly compare the series given to us in the question with a π-series. First, we have the sum from π equals one to β of seven π cubed divided by five π to the fourth power. We can see our numerator and our denominator share a factor of π cubed. So weβll cancel this shared factor of π cubed in our numerator and our denominator. This gives us the sum from π equals one to β of seven divided by five times π. We might not see how we can directly compare this with a π-series. However, remember that π is actually equal to π raised to the first power.

We can now see this is very similar to a π-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 π 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 π equals one to β of one divided by π to the first power. This is seven-fifths times a series. In fact, this series is a π-series where π is equal to one. And we know that a π-series when π is equal to one must be divergent.

So what weβve 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 π-series test, we were able to show the sum from π equals one to β of seven π cubed divided by five π to the fourth power is divergent.