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so as we talked about in the last video we've seen many examples so starting with a geometric series expanded out and then assuming that it's common ratio that the absolute value of the common ratio is less than one finding what the sum of that might be and we've proven with this formula in previous videos but now let's go the other way around let's try to take some function let's say H of X being equal to 1 over 3 plus x squared and let's try to put it in this form and then once we put it in that form we can think about what a and our common ratio is and then try to represent it as a act for represented as an actual geometric series so I encourage you to pause the video and try to do that right now so let's see the first thing that you might notice is we have a 1 here instead of a 3 so let's try to factor out a 3 so this is equal to 1 over 3 times 1 plus x squared over 3 and now we can since we don't want that 3 in the denominator we can think about this as 1 over 3 so we could say this is 1/3 / / / let me do in that purple color 1/3 / 1 and we don't want to just add some we want to subtract our common ratio so 1 - and let me write our common ratio here in yellow 1 minus negative x squared over 3 so now we've written this in that form and so now we could say we could say that the sum let me write it here in let me do it in a new color so let me do it in blue so now we could say that the sum from N equals 0 to infinity of C our first term is 1/3 1/3 times our common ratio so our common ratio to the nth power common ratio is negative negative x squared over 3 and if we wanted to expand this out this would be equal to so the first term is 1 third times all of this to the 0th power so it's just going to be 1/3 and so each successive term is is going to be the previous term times our common ratio so one-third times negative x squared over three is going to be negative we write as negative 1/9 x squared right to go from that to that you have to multiply by let's see one third to negative one third you have to multiply by negative one third and you have we multiplied by x squared as well and our our next term we're going to multiply by negative x squared over three again so it's going to be plus it's going to be plus the negative times a negative is a positive plus one over twenty-seven x to the fourth x squared times x squared X to X to the fourth power and we just keep going on and on and on and when when X when the when our or I should say when this converges so over the interval of convergence this is going to converge to H of X now what is the interval of convergence here and I encourage you to pause the video and think about it well the interval of convergence is the interval over which your common ratio the absolute value of your common ratio is less than one so let me write this right over here so your our absolute value of negative x squared over three the absolute value of negative x squared over three has to be less than one well the absolute value this this is going to be a negative number this is the same thing as saying let me scroll down a little bit this is the same thing as saying that the absolute value of x squared over three has to be less has to be has to be less than one and this is another way of saying well one thing that you might that might jump out at you is that x squared this is going to be positive no matter what or I guess I should say this is going to be non-negative no matter what so this is another way of saying that x squared x squared over three has to be less than one right I don't want to confuse you in this step right over here but the absolute value of x squared over three is just going to be x squared over three because this is never going to take on a negative value and so we can multiply both sides by three I'll go up here now to do it multiply both sides by three to say that x squared needs to be less than three and so that means that means that the absolute value of x needs to be less than the square root of three or we could say that X is greater than the negative square root of three and it is less than the square root of three so this is the interval of convergence this is the interval interval of convergence of convergence for this series for this power series it's a geometric series which is a special case of a power series and over the interval over the interval of convergence that is going to be equal to its going to be equal to 1 over 3 plus x squared so as long as X is in this interval it's going to take on the same values as our original function which is a pretty neat idea

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