### Video Transcript

For the series which is equal to the sum from π equals one to β of one divided by five to the πth power minus two to the πth power, use the limit comparison test to determine whether the series converges or diverges.

The question tells us to use the limit comparison test. So letβs recall that the limit comparison test tells us that if we have a sequence π π which is greater than or equal to zero and a sequence π π which is greater than zero, where π is greater than or equal to one. And that the limit as π approaches β of the quotient π π over π π is equal to π, where π is a finite positive number. Then we can conclude that the sum from π equals one to β of π π and the sum from π equals one to β of π π both converge or both diverge.

The concluding statement in the limit comparison test tells us that the convergence and divergence of these two series are the same. So if we can find sequences π π and π π such that the prerequisites of the limit comparison test are true and where we can determine where one of the series either converges or diverges. Then we can use the limit comparison test to determine whether the other series converges or diverges.

Since the question is asking us about the sum from π equals one to β of one divided by five to the πth power minus two to the πth power. By using the limit comparison test, we should set however π π or π π to be equal to one divided by five to the πth power minus two to the πth power. Letβs try setting this equal to be π. We will see why shortly. Now, letβs see what expression we get for our limit of our quotient π π over π π, where we have that this limit is equal to the limit as π approaches β of π π multiplied by the reciprocal of π π. And we know the reciprocal of π π is just five to the πth power minus two to the πth power. This gives us that the limit of our quotient is equal to the limit as π approaches β of π π multiplied by five to the πth power minus two to the πth power. And we want this to be equal to π, where π is some finite positive number.

We see if we just take the limit of five to the πth power minus two to the πth power, then this gets bigger and bigger as π gets bigger and bigger. Because five is greater than two, so this limit is to β. Since we want the limit of this product to be some finite positive number, we can conclude that the limit of π π as π approaches β must be equal to zero. At this point, letβs let π π be equal to one divided by π π and substitute this into our limit. This gives us that we want the limit as π approaches β of five to the πth power minus two to the πth power divided by our new sequence π π to be equal to some finite positive number.

One thing we could try is setting our sequence π π to be equal to five to the πth power because five to the πth power is the fastest growing part of our numerator. We can then split this fraction so that weβre dividing both terms in the numerator by five to the πth power. And we know that five to the πth power divided by five to the πth power is just equal to one. Then we can use the fact that π₯ to the πth power divided by π¦ to the πth power is equal to π₯ over π¦ all raised to the πth power. To see that two to the πth power divided by five to the πth power is just two-fifths raised to the πth power. This gives us the limit as π approaches β of one minus two-fifths to the πth power.

Now, we know that the limit as π approach β of π to the πth power is equal to zero when the size of π is less than one. So we can use this to conclude that the limit of two-fifths to the πth power as π approach β is equal to zero. Therefore, since the limit of one as π approaches β is just equal to one, we have that the limit of π π over π π as π approaches β is just equal to one. Since one is a finite positive number, we have shown one of the prerequisites for our limit comparison test.

Next, letβs check if the sequence π π is nonnegative for π greater than or equal to one. We have that the sequence π π is equal to one over five to the πth power which we can rewrite as one-fifth to the πth power. Since π is a positive integer, one-fifth to the πth power is just one-fifth multiplied by itself π times. And we know that one-fifth is a positive number. So multiplying π positive numbers together is always going to give us a positive number. So this is strictly greater than zero. So we have shown that our sequence π π is nonnegative for π greater than or equal to one.

The last thing we need to show before we use our limit comparison test is that the sequence π π is positive for π greater than or equal to one. We have our sequence π π is equal to one divided by five to the πth power minus two to the πth power. And when π is greater than or equal to one, we know that five to the πth power is bigger than two to the power. So subtracting two to the πth power from both sides of that inequality gives us five to the πth power minus two to the πth power is positive. Therefore, what we have shown is that our sequence π π is equal to one divided by a positive number. And therefore, it must be greater zero. Therefore, weβve shown all of our prerequisites to be true. So we can now use the limit comparison test.

The limit comparison test tells us that whether or not the series in our question diverges or converges is equivalent to whether the sum from one to β of one divided by five to the πth power converges or diverges. We see that this is a geometric series. And we know that for geometric series, the sum from π equals one to β of π to the πth power is equal to π divided by one minus π when the size of π is less than one. Therefore, since we know that our series is equal to the sum from π equals one to β of one-fifth to the πth power, we can conclude that it is also equal to one-fifth divided by one minus one-fifth. Which, if we evaluate, we see is equal to a quarter.

Therefore, since that series converged, we can conclude, by using the limit comparison test, that the sum from π equals one to β of one divided by five to the πth power minus two to the πth power must converge.