Power series
Imagine you have a jar of candies and you want to count how many candies are in it. But instead of counting one by one, you want to use a formula to find out the total number of candies.
A power series is like a magical formula that helps you do this kind of counting. It's a way of adding up an infinite number of terms, just like adding up an infinite number of candies in the jar.
In math terms, a power series is a formula that looks like this:
f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + ...
Each term in the formula has a coefficient (a_0, a_1, a_2, etc.) and a power of x (x, x^2, x^3, etc.). The first term (a_0) is just a regular number, while the other terms have x raised to different powers.
Here's how you use a power series to count candies. Let's say you have 3 candies in the jar, and you want to find out how many candies you'll have if you add 2 more. You can use a power series with x=2 and a_0=3 to do this:
f(x=2) = a_0 + a_1x + a_2x^2 + a_3x^3 + ...
f(x=2) = 3 + a_1(2) + a_2(2^2) + a_3(2^3) + ...
Now, you might not know what the other coefficients are (a_1, a_2, a_3, etc.). But if you're lucky, you might be able to figure out a formula for them. For example, let's say you notice that each coefficient is twice the previous one:
a_1 = 2a_0
a_2 = 2a_1
a_3 = 2a_2
...
If you substitute these formulas into the power series formula, you get:
f(x=2) = 3 + 2a_0(2) + 2a_0(2^2) + 2a_0(2^3) + ...
Now you can simplify this by factoring out 2a_0:
f(x=2) = 3 + 2a_0(1 + 2 + 2^2 + 2^3 + ...)
The sum inside the parentheses is called a geometric series, and it's a special type of series that has a nice formula:
1 + r + r^2 + r^3 + ... = 1 / (1 - r)
So in this case, the sum is:
1 + 2 + 2^2 + 2^3 + ... = 1 / (1 - 2) = -1
Yes, the sum is negative, but don't worry about that for now. Just substitute it back into the power series formula:
f(x=2) = 3 + 2a_0(-1) = 3 - 2a_0
And since a_0 was 3 (the number of candies we started with), we get:
f(x=2) = 3 - 2(3) = -3
Uh oh, that's not right! It looks like our formula is giving us a negative number of candies. What went wrong?
The problem is that the geometric series formula only works if the absolute value of r is less than 1. In our case, r is 2, which is greater than 1. So the formula doesn't apply.
But that's okay! Power series are often used to approximate functions, rather than give exact values. So in this case, we could say that adding 2 candies to a jar with 3 candies might give us roughly -3 candies (which doesn't make sense in the real world, but we're just doing math here).
Overall, power series are a powerful tool for solving math problems and approximating functions. They allow you to add up an infinite number of terms, and can be used to model many different types of phenomena, from the number of candies in a jar to the behavior of electrons in a quantum mechanical system.
A power series is like a magical formula that helps you do this kind of counting. It's a way of adding up an infinite number of terms, just like adding up an infinite number of candies in the jar.
In math terms, a power series is a formula that looks like this:
f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + ...
Each term in the formula has a coefficient (a_0, a_1, a_2, etc.) and a power of x (x, x^2, x^3, etc.). The first term (a_0) is just a regular number, while the other terms have x raised to different powers.
Here's how you use a power series to count candies. Let's say you have 3 candies in the jar, and you want to find out how many candies you'll have if you add 2 more. You can use a power series with x=2 and a_0=3 to do this:
f(x=2) = a_0 + a_1x + a_2x^2 + a_3x^3 + ...
f(x=2) = 3 + a_1(2) + a_2(2^2) + a_3(2^3) + ...
Now, you might not know what the other coefficients are (a_1, a_2, a_3, etc.). But if you're lucky, you might be able to figure out a formula for them. For example, let's say you notice that each coefficient is twice the previous one:
a_1 = 2a_0
a_2 = 2a_1
a_3 = 2a_2
...
If you substitute these formulas into the power series formula, you get:
f(x=2) = 3 + 2a_0(2) + 2a_0(2^2) + 2a_0(2^3) + ...
Now you can simplify this by factoring out 2a_0:
f(x=2) = 3 + 2a_0(1 + 2 + 2^2 + 2^3 + ...)
The sum inside the parentheses is called a geometric series, and it's a special type of series that has a nice formula:
1 + r + r^2 + r^3 + ... = 1 / (1 - r)
So in this case, the sum is:
1 + 2 + 2^2 + 2^3 + ... = 1 / (1 - 2) = -1
Yes, the sum is negative, but don't worry about that for now. Just substitute it back into the power series formula:
f(x=2) = 3 + 2a_0(-1) = 3 - 2a_0
And since a_0 was 3 (the number of candies we started with), we get:
f(x=2) = 3 - 2(3) = -3
Uh oh, that's not right! It looks like our formula is giving us a negative number of candies. What went wrong?
The problem is that the geometric series formula only works if the absolute value of r is less than 1. In our case, r is 2, which is greater than 1. So the formula doesn't apply.
But that's okay! Power series are often used to approximate functions, rather than give exact values. So in this case, we could say that adding 2 candies to a jar with 3 candies might give us roughly -3 candies (which doesn't make sense in the real world, but we're just doing math here).
Overall, power series are a powerful tool for solving math problems and approximating functions. They allow you to add up an infinite number of terms, and can be used to model many different types of phenomena, from the number of candies in a jar to the behavior of electrons in a quantum mechanical system.