![]() And if this looks unfamiliar to you, I encourage you to watch the video where we find the formula, This is going to be equal to our first term which is a over one minus our common ratio, one minus our common ratio. ![]() K equal zero to infinity and you have your first term a times r to the k power, r to the k power, assuming this converges, so, assuming that the absolute value of your common ratio is less than one, this is what needs toīe true for convergence, this is going to be equal to, We prove it in other videos, if you have a sum from So now that we've seen that we can write a geometric series in multiple So these are allĭescribing the same thing. And so when k is equal to two, that is this term right over here. That's gonna be eight timesġ/3 to the first power. When k is equal to one, that's gonna be our second term here. You get eight times 1/3 to the zero power, which is indeed eight. Should be the first term right over here. Just as a reality check, and I encourage you to do the same. Times our common ratio, times our common ratio 1/3 to the k power. What's our first term? Our first term is eight. And this is an infinite series right here, we're just gonna keep on going forever, so to infinity of, well And we could start at zero or at one, depending on how we like to do it. This, the first thing we wrote is equal to this, which is equal to, this is equal to the sum. It this way, you're like, okay, we could write So we could rewrite the series as eight plus eight times 1/3, eight times 1/3, plus eight times 1/3 squared, eight times 1/3 squared. Then go to the next term we are going to multiply by 1/3 again, and we're going to keep doing that. So let's see, to go from theįirst term to the second term we multiply by 1/3, And just to make sure that we're dealing with a geometric series, let's make sure we have a common ratio. The distance travelled by a bouncing ball is a classic application of this concept.Some practice taking sums of infinite geometric series.There are some real-life applications among the problems on this exercise.The concepts in this exercise show up in second semester calculus as related to Taylor polynomials.Summation (or sigma) notation is a notation used for representing long sums. ![]()
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