 In this video, we provide the solution to question number four for practice exam number four for math 12-20. We're given the following series, take the sum as n ranges from one to infinity of the terms two over five to the n. We have to determine if this thing is convergent. If it's convergent, we find the sum and if it's divergent, we would just select divergent, of course. So when I look at that series here, it is in fact a geometric series. I can notice that because as the numerator is constant, the denominator is growing exponentially. So five to the n there. That indicates we have a geometric series. So the convergence of a geometric series comes down to its common ratio. The common ratio here is gonna be one-fifth. And you'll notice that the absolute value of the ratio, which is one-fifth, is in fact strictly less than one. So it is in fact convergent. That means f is not the correct answer. We now need to compute what is the value then. So given a geometric series here is two over five to the n. The formula we have is that on the top, you're gonna get the very first term of the sequence. So if you plug one into the sequence there, you're gonna get two-fifths. That's the first term of the sum. Then in the denominator, you get one minus the ratio, which the ratio is one-fifth. Simplifying this, we get two-fifths. Well, actually I'm gonna save myself a little bit of effort here. I'm just gonna times the top and bottom by five to clean up these fractions here. Because in the top, you get two-fifths times five, which is going to be two. In the denominator, you distribute the five. You're gonna get five minus one. So this gives me two over four, which gives me one-half. So we see that the correct answer is that this series is convergent and the sum is equal to one-half.