# Video: Geometric Interpretation of Multiplication by π

Let us consider π(π₯) = π₯/(3 β π₯). Find a power series representation for π(π₯). Find the radius of convergence.

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### Video Transcript

Let us consider the function π of π₯ is equal to π₯ divided by three minus π₯. Find a power series representation for the function π of π₯. Find the radius of convergence.

The question gives us a rational function π of π₯. It wants us to find the power series representation of the function π of π₯. And it wants us to then find the radius of convergence for this power series. To do this, we recall a fact about the sum of an infinite geometric series. The sum from π equal zero to β of π multiplied by π to the πth power is equal to π divided by one minus π, when the absolute value of our ratio π is less than one.

So if we can write our rational function π of π₯ in the form π divided by one minus π, then we can use this fact about geometric series to write it as a power series with a radius of convergence the absolute value of π is less than one. Weβll start by noticing that we can factor a coefficient of π₯ outside of our rational function. This gives us π₯ multiplied by one over three minus π₯. We want our denominator to be in the form one minus the ratio π. However, we have three minus π₯ as our denominator. To have a denominator in the form one minus π, weβre going to factor out a coefficient of three in our denominator. This gives us three multiplied by one minus π₯ divided by three. We can then simplify this by taking the coefficient of three in our denominator and putting it outside of our rational function.

If we then set π equal to one and π equal to π₯ divided by three, we can use our infinite geometric series to rewrite our expression. Using our infinite geometric series formula, we have that π of π₯ is equal to π₯ divided by three multiplied by the sum from π equals zero to β of one multiplied by π₯ over three to the πth power, when the absolute value of our ratio π₯ divided by three is less than one. We can simplify this further.

First, we can remove the multiplication by one. Next, we can take our coefficient of π₯ divided by three inside of our sum to give us π₯ over three multiplied by π₯ over three to the πth power. However, we can simplify this further to just be π₯ over three all raised to the power of π plus one. This gives us that our function π of π₯ is equal to the sum from π equals zero to β of π₯ over three to the power of π plus one, when the absolute value of π₯ over three is less than one.

Next, the question wants us to find the radius of convergence of this power series. We recall that we called π the radius of convergence of a power series if the power series converges when the absolute value of π₯ is less than π and diverges when the absolute value of π₯ is greater than π. Unless the power series converges for all values of π₯, in which case we call the radius of convergence β. We recall that for a geometric series, the sum from π equals zero to β of π multiplied by π to the πth power will diverge whenever the absolute value of our ratio π is greater than one. In our case, our ratio π is π₯ divided by three. So our power series converges when the absolute value of π₯ over three is less than one and diverges when the absolute value of π₯ over three is greater than one.

We can then rearrange both of these expressions. The absolute value of π₯ over three being less than one is the same as saying the absolute value of π₯ is less than three. And if the absolute value of π₯ over three is greater than three, this is the same as saying the absolute value of π₯ is greater than three. So our power series converges when the absolute value of π₯ is less than three and diverges when the absolute value of π₯ is greater than three. Therefore, our radius of convergence π is equal to three. And our power series for π of π₯ will converge whenever the absolute value of π₯ is less than three.