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Main content
Current time:0:00Total duration:6:49
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Video transcript

we've already seen many examples of infinite series but what's exciting about we're about to do in this video is we're going to use infinite series to define a function and the most common one that you will see in your mathematical careers is the power series this is the power series and I'm about to write a general case of the power series so I could imagine a function f of X being defined as the infinite sum so going from N equals zero to infinity of a sub n so a sub n is just going to be the coefficient on each term times our variable X minus some constant C you can almost imagine this is shifting our function to the nth power so if I were to expand this out I have my first terms coefficient a sub zero times X minus C to the 0th power plus a sub one times X minus C to the first power this one of course will simplify to just a sub zero this would simplify to a sub one times X minus C plus a sub two times X minus C squared and I could just keep going on and on and on now when you see this you might say well our geometric series don't look like a special case of a power series if our common ratio was an X instead of an R in that case or if our if our if we if our common ratio was a variable I guess I could say and you would be right that absolutely would be the case so a geometric series so let's just think about defining a function in terms of a geometric series and of course we don't have to use X all the time as the independent variable but this is kind of the most difficult convention I guess we could also use R as in the dependent variable if we want it as well but imagine let's imagine a function G of X we could have G of R if we wanted but let's do G of X is equal to the sum from N equals 0 to infinity of a times X to the N so this is kind of a this is kind of a typical geometric series here and what's the difference between this and this well the difference is is here for every term we're going to have the same coefficient a while over here we have a sub n we're multiplying by a different thing every time up here we're multiplying by the same anything over here and in this case this particular geometric series I just made instead of having X minus C to the end we have just X to the end 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 times X to the 0 which is just going to be a plus a times X to the first plus a times x squared and we just go on and on and on forever now what's exciting about this is we know we know under what we know that this under certain conditions this will actually give us a finite value this will actually converge this will actually I guess may give us a sensible answer so under what conditions does that happen well this converges is if each of these terms gets smaller and smaller and smaller and each of these terms get smaller and smaller smaller if the absolute value of our common ratio the absolute value of our common ratio is less than 1 so let me write that down so this converges converges if the absolute value of our common ratio is less than 1 or another way of thinking about it this is another way of saying this is another way of saying is that X is in the interval between it's less than 1 and it is greater than negative 1 and this term right over here now X is a variable X can vary between those values we're defined a function in terms of X we call this the interval of convergence in interval interval of convergence and so we know that if X is in this interval this is going to give us a finite sum 1 we know what that finite sum is it's going to be it's going to be equal to if it converges so if it converges this is going to be equal to our first term which is just a this simplifies to a right over here over 1 minus our common ratio what's our common ratio our common ratio in this example is X going from one term to the next we're just multiplying by X we're just multiplying by X right over there now this is pretty neat because we're going to be able to use this fact to put more traditionally defined functions into this form and then try to expand them out using a geometric series and this whole idea of using power series or in the special case geometric series to represent functions has all sorts of applications in engineering and finance using a finite number of terms of these of these series you can kind of approximate the functions in a in a way that's simpler for the human brain to understand or maybe a simpler way to manipulate in some way but what's interesting here is instead of just going from the sum to instead of going from this expanded out version to this this kind of finite value we're now going to start being able to take something in this form and expand it out into a geometric series but we have to be careful to make sure that we're only doing it over the interval of convergence this is only going to be true over the interval of convergence now one other term you might see in your mathematical career is radius radius of convergence convergence and this is how far up to what value but not including this value so if as long as our as long as our x value strays no stays less than a certain amount from our C value then this thing will converge now in this case our C value is zero so our art so we could ask ourselves a question of what is how as long as X stays within some value of zero this thing is going to converge well you see it right over here as long as X stays within one of zero so as long as it can get it can't go all the way to one but as long as it stays less than one or as long as it stays greater than negative one so it can strain no more or I guess it should it has to it can it can straight anything less than one away from zero either in the positive direction or the negative direction then this thing will still converge so we could say that our radius of convergence our radius of convergence is equal to 1 another way to think about it our interval of convergence we're going from negative 1 to 1 not including those two boundaries so our interval is 2 so our radius of convergence is half of that as long as we stay within 1 of 0 R as long as X stay within one of 0 and that's the same thing as saying this right over here this series is going to converge