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### Course: Integral Calculus>Unit 5

Lesson 13: Power series intro

# Power series intro

Power series is a sum of terms of the general form aₙ(x-a)ⁿ. Whether the series converges or diverges, and the value it converges to, depend on the chosen x-value, which makes power series a function. Created by Sal Khan.

## Want to join the conversation?

• How are there any applications for this? A function, whose x-values are restricted between -1 and 1 doesnt seem to be very useful?
• A couple points on that:

1. Not all functions have such a small radius of convergence. The power series for sin(x), for example, converges for all real values of x. 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).

2. Even for functions with small radii of convergence, power series still give us the ability to calculate values that would otherwise be unapproachable. 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.

3. We can just shift the center of our power series if we want to approximate a value outside the interval of convergence. 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.

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. That gives us, among other things, a way to program machines to calculate values of functions like sin(x) and sqrt(x).

Hope that helps.
• 1.) Is a polynomial a type of power series where a_n is 0 for certain n (it has to be 0 because a polynomial has a finite number of terms)?

2.) Could the geometric series be generalized to the summation from n=0 to Infinity of a*(x-c)^n?
• 1) Yes, polynomials could be said to be special cases of the power series.

2) Sure. The new convergence condition would be that |x - c| < 1.
• Hey guys, I had trouble of understanding what Power Series is. What is this power series for? is it just expanding a given function or something? your reply will be helpful thanks!
• A power series is simply a category of series. If a series follows that specific form, then it's called a power series.
(1 vote)
• This tells how to find the interval of convergence for a geometric series but doesn't explain how to do so for a power series, which is what i thought this video was supposed to be about......
• It was subtle and easy to miss, but at Sal mentions that Geometric Series is a special case of the Power Series where the common ratio is an x, rather than an r. This is important, he is saying that geometric series, while you may not have thought about them as power series, or even as a representation of a function, they are, and that when you analyse a geometric series, it is just a special case of a power series. In fact, this is the usual method of introducing power series, by developing the concept of a geometric series, and applying that when meeting the power series with the "good news" that the general formula a/(1-r) becomes a/(1-x) and can be used to determine the radius of convergence.
The following videos will make this point clearer.
• At , Sal says that the first term in the series will simplify to a_0. Is that still true when x = c? Wasn't 0^0 an indeterminate form?
• You are correct that 0^0 is an indeterminate form! Three things that come to mind:
1) the limit as x --> 0 for x^0 is 1.
2) 0^0 does not exist, because it is indeterminate (only the limit exists)
3) if x-c = 0 then all of the rest of the series terms would go to zero
So if I had to make an educated guess, I would say that there are two possible outcomes:
1) we start our summation at n = 0 and then the series fails, because it contains an indeterminate form
2) we start our summation at n = 1 and then the series = 0
• Whats the difference between a power series as a function of x and a geometric series as a function of x?
• A geometric series is simply a special case of a power series, where c = 0 and aₙ = a.
• At around the mark, if the interval of convergence included 1 or -1, would the radius of convergence still be 1?
• To determine the radius of convergence, do not worry about whether the endpoints are included or not. The radius of convergence would be one regardless of whether or not the endpoints were included.
• So is radius always half the difference in values for the interval?
• Yes, assuming the absolute value of the values are equal. For example if the interval of convergence is -2 < x < 1 and c = 0, the radius of convergence is actually 1, since the series converges as long as x is within 1 of 0. The series does not converge as long as x is within 2 of 0, since x = 2 will not converge.
(1 vote)
• I'm confused how it simplifies to a/(1-x) because if you run it with say x = .5 it doesn't work
• This is actually a result of adding a geometric sequence if you add a sequence to infinity where there is a common ration between each term and the absolute value of that ratio is less than one then that series converges and it equals a/1-r where r is the common ratio and a is the first term I suggest watching these videos to refresh your memory about geometric series

I don't understand what led you to believe that it doesn't work when x=0.5 if you set x=0.5 you get 2a which makes sense remember that we are adding a + a/2 + a/4 + a/8 .. to inf.
if you take a common then you will have a * ( 1 + 1/2 + 1/4 + 1/8 .... )
there are beautiful visualization why the right term results in two and so you would have a*2
which doesn't seem wrong to me.
here is a visualization for the previous result hope this helps and keep studying and good luck.
• What is center of convergence specifically?
• The center of convergence is where the distance from the lowest point to a specific number(the center) is the same as the distance from the highest point to a specific number(the center). Another word for the distance is the radius of convergence. Example: the center of convergence of the interval -1<x<1 is 0, because the radius is 1. You can find the center by subtracting the bigger number of the interval by the smaller number of the interval and then dividing by 2. Example: Center of -1<x<1= (1-(-1))/2=2/2=1.