What's our common ratio? Our common ratio in Right over here- over 1 minus our common ratio. Term, which is just a- this simplifies to a So this converges ifīetween- it's less than 1 and it is greater Smaller and smaller and smaller if the absolute value of ourĬommon ratio is less than 1.
#CAUCHY SEQUENCE KHAN PLUS#
X to the 0, which is just going to be a, plusĪ times x to the first, plus a times x squared. This is a special case when c is equal to 0. Having x minus c to the n, we have just x to the n.
#CAUCHY SEQUENCE KHAN SERIES#
Particular geometric series I just made, instead of Same coefficient a, while over here we have a sub n. Here, for every term we're going to have the And what's the differenceīetween this and this? Well, the difference is is If we wanted, but let's do g of x is equal I guess we could also use rĪs an independent variable if we wanted as well. Independent variable, but this is kind of the So let's just think aboutĭefining a function in terms of a geometric series. Variable, I guess I could say? And you would be right. Instead of an r in that case, or if our common ratio was a This, you might say, aren't our geometricĬase of a power series if our common ratio was an x This would simplify to a subġ times x minus c plus a sub 2 times x minus c squared. To the 0-th power, plus a sub 1 times x minusĬ to the first power. Imagine this is shifting our function to the n-th power.
The coefficient on each term- times our variable x To infinity of a sub n- so a sub n is just going to be And I'm about to write a generalĬase of the power series. One that you will see in your mathematicalĬareers is the power series. We're about to do in this video is we're going to use infinite That gives us, among other things, a way to program machines to calculate values of functions like sin(x) and sqrt(x).Įxamples of infinite series. In short, power series offer a way to calculate the values of functions that transcend addition, subtraction, multiplication, and division - and they let us do that using only those four operations. For example, to calculate ln(5), we could use a power series for ln(x) centered at x=e^2 instead of x=1, which would put x=5 inside our interval of convergence. We can just shift the center of our power series if we want to approximate a value outside the interval of convergence. The series for ln(x) centered at x=1 converges only over a radius of 1, but for calculating a number like ln(0.36), it's obviously still useful.ģ. Even for functions with small radii of convergence, power series still give us the ability to calculate values that would otherwise be unapproachable. That gives you a way to calculate sin(x) for any value using nothing but a polynomial, which is an extremely powerful concept (especially given that we can't just evaluate a number like sin(47) because 47 doesn't fit nicely with the periodicity of the sine function we have to use some other method that we can actually calculate, which generally boils down to polynomial approximations - power series).Ģ. The power series for sin(x), for example, converges for all real values of x. Not all functions have such a small radius of convergence.
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