Video transcript
We've already seen manyexamples of infinite series. But what's exciting about whatwe're about to do in this video is we're going to use infiniteseries to define a function. And the most commonone that you will see in your mathematicalcareers is the power series. And I'm about to write a generalcase of the power series. So I could imaginea function, f of x, being defined asthe infinite sum. So going from n equals 0to infinity of a sub n-- so a sub n is just going to bethe coefficient on each term-- times our variable xminus some constant c. You could almostimagine this is shifting our function to the n-th power. So if I were toexpand this out, I have my first term'scoefficient, a sub 0, times x minus cto the 0-th power, plus a sub 1 times x minusc to the first power. This one, of course, willsimplify just a sub 0. This would simplify to a sub1 times x minus c plus a sub 2 times x minus c squared. And I could just keepgoing on and on and on. Now, when you seethis, you might say, aren't our geometricseries, don't those look like a specialcase of a power series if our common ratio was an xinstead of an r in that case, or if our common ratio was avariable, I guess I could say? And you would be right. That absolutelywould be the case. So a geometric series. So let's just think aboutdefining a function in terms of a geometric series. And of course, wedon't have to use x all the time as theindependent variable, but this is kind of themost typical convention. I guess we could also use ras an independent variable if we wanted as well. But let's imaginea function g of x. We could have g of rif we wanted, but let's do g of x is equalto the sum from n equals 0 to infinityof a times x to the n. So this is kind of a typicalgeometric series here. And what's the differencebetween this and this? Well, the difference is ishere, for every term we're going to have thesame coefficient a, while over here we have a sub n. We're multiplying by a differentthing every time up here. We're multiplying by thesame thing over here. And in this case, thisparticular geometric series I just made, instead ofhaving x minus c to the n, we have just x to the n. So you could say, well,this is a special case when c is equal to 0. And we can expand it out. This is a timesx to the 0, which is just going to be a, plusa times x to the first, plus a times x squared. And we just go on andon and on forever. Now, what's excitingabout this is we know that this, undercertain conditions, will actually giveus a finite value. This will actually converge. This will actually, I guess,give us a sensical answer. So under what conditionsdoes that happen? Well, this convergesif each of these terms gets smaller andsmaller and smaller. And each of these terms getssmaller and smaller and smaller if the absolute value of ourcommon ratio is less than 1. So let me write that down. So this converges ifthe absolute value of our common ratiois less than 1. Or another way ofthinking about it, this is another wayof saying that x is in the intervalbetween-- it's less than 1 and it is greaterthan negative 1. And this term right overhere, now x is a variable. x can vary between those values. We're defining afunction in terms of x. We call this theinterval of convergence. And so we know that ifx is in this interval, this is going togive us a finite sum. And we know whatthat finite sum is. It's going to be equalto-- if it converges. So if it converges,this is going to be equal to our firstterm, which is just a-- this simplifies to aright over here-- over 1 minus our common ratio. What's our common ratio? Our common ratio inthis example is x. Going from one term to the next,we're just multiplying by x. We're just multiplyingby x right over there. Now, this is prettyneat, because we're going to be ableto use this fact to put moretraditionally-defined functions into this form, andthen try to expand them out using a geometric series. And this whole ideaof using power series, or in this specialcase, geometric series to represent functions,has all sorts of applications inengineering and finance. Using a finite number ofterms of these series, you could kind ofapproximate the functions in a way that's simplerfor the human brain to understand, ormaybe a simpler way to manipulate in some way. But what's interestinghere is instead of just going from the sumto-- instead of going from this expanded-out versionto this kind of finite value, we're now going to startbeing able to take something in this form and expand itout into a geometric series. But we have to becareful to make sure that we're only doing it overthe interval of convergence. This is only going to betrue over the interval of convergence. Now, one other term you mightsee in your mathematical career is a radius. Radius of convergence. And this is how far--up to what value, but not including this value. So as long as our x value staysless than a certain amount from our c value, thenthis thing will converge. Now in this case,our c value is 0. So we could askourselves a question. As long as x stayswithin some value of 0, this thing is going to converge. Well, you see itright over here. As long as x stayswithin one of 0. It can't go all theway to 1, but as long as it stays lessthan 1, or as long as it stays greaterthan negative 1. It can stray anythingless than one away from 0, either inthe positive direction or the negative direction. Then this thingwill still converge. So we could say that our radiusof convergence is equal to 1. Another way to think about it,our interval of convergence-- we're going fromnegative 1 to 1, not including thosetwo boundaries, so our interval is 2. So our radius ofconvergence is half of that. As long as x stayswithin one of 0, and that's the same thing assaying this right over here, this series isgoing to converge.
