Video: Finding the Maclaurin Series of the Product of a Power Function and an Exponential Function

Video Transcript

Find the Maclaurin series of 𝑥 cubed times 𝑒 to the power of two 𝑥.

We need to find the Maclaurin series of 𝑥 cubed times 𝑒 to the power of two 𝑥. And there’s a couple of different ways we could do this. For example, we could directly use the definition of a Maclaurin series. And to do this, we would need to find derivatives of 𝑥 cubed times 𝑒 to the power of two 𝑥, and we could do this by using the product rule. And this would work; however, this is a complicated method. We would need to find multiple derivatives of the product of two functions.

So, instead, we need to notice something. We have 𝑥 cubed multiplied by 𝑒 to the power of two 𝑥. And we already know how to find the Maclaurin series of 𝑒 to the power of two 𝑥. Then, we could just multiply this through by 𝑥 cubed. So, let’s start by finding the Maclaurin series of 𝑒 to the power of two 𝑥. Once again, we could do this directly from the definition of Maclaurin series. However, we can also do this by using the Maclaurin series for 𝑒 to the power of 𝑥. And in fact, this method is a lot easier.

We need to start by recalling that 𝑒 to the power of 𝑥 will be equal to the sum from 𝑛 equals zero to ∞ of 𝑥 to the 𝑛th power divided by 𝑛 factorial. And it’s worth pointing out we know this power series converges for any real value of 𝑥. We want to use this to find the Maclaurin series for 𝑒 to the power of two 𝑥. To do this, all we need to do is replace 𝑥 with two 𝑥. Doing this, we get 𝑒 to the power of two 𝑥 is equal to the sum from 𝑛 equals zero to ∞ of two 𝑥 all raised to the 𝑛th power divided by 𝑛 factorial.

And it’s worth pointing out since our original power series converges for any real values of 𝑥, our new power series will also converge for any real value of 𝑥. And we can simplify this expression slightly. We can distribute the exponent over our parentheses. Two 𝑥 all raised to the 𝑛th power is equal to two to the 𝑛th power times 𝑥 to the 𝑛th power. So, now, we found the Maclaurin series expansion for 𝑒 to the power of two 𝑥. All we need to do is multiply both sides of this equation through by 𝑥 cubed.

Multiplying both sides of this equation through by 𝑥 cubed, we get 𝑥 cubed times 𝑒 to the power of two 𝑥 is equal to 𝑥 cubed times the sum from 𝑛 equals zero to ∞ of two to the 𝑛th power over 𝑛 factorial multiplied by 𝑥 to the 𝑛th power. And we can simplify this expression slightly. 𝑥 cubed is not varying as our value of 𝑛 varies, so we can bring this factor of 𝑥 cubed inside of our sum. Then, we have one more piece of simplification we can do. 𝑥 to the 𝑛th power multiplied by 𝑥 cubed is equal to 𝑥 to the power of 𝑛 plus three, and this gives us our final answer.

Therefore, by using what we knew about Maclaurin series and the Maclaurin series for the exponential function, we were able to show the Maclaurin series of 𝑥 cubed times 𝑒 to the power of two 𝑥 is equal to the sum from 𝑛 equals zero to ∞ of two to the 𝑛th power divided by 𝑛 factorial multiplied by 𝑥 to the power of 𝑛 plus three.