Question Video: Finding the Maclaurin Series of a Rational Function | Nagwa Question Video: Finding the Maclaurin Series of a Rational Function | Nagwa

# Question Video: Finding the Maclaurin Series of a Rational Function Mathematics • Higher Education

Find the Maclaurin series for π(π₯) = 1/(10π₯ + 1).

02:46

### Video Transcript

Find the Maclaurin series for π of π₯ is equal to one divided by 10π₯ plus one.

We need to find the Maclaurin series for the function π of π₯. Thereβs a few different ways of doing this. We might be tempted to use the definition of a Maclaurin series. And we recall the Maclaurin series for π of π₯ is the sum from π equals zero to β of the πth derivative of π of π₯ with respect to π₯ evaluated at zero divided by π factorial multiplied by π₯ to the πth power. And this will only be valid for values of π₯ where our series is convergent. And we see, to find this power series, we need to find the πth derivative of π of π₯ with respect to π₯ evaluated at zero.

In other words, if we wanted to do this directly from our definition, we would need to differentiate our function π of π₯ π times. And in fact, this would work. We could do this by using the quotient rule, the chain rule, or the general power rule. But thatβs quite a complicated method; itβs very easy to make a mistake. In fact, thereβs an easier method we could use. Instead, remember, we can find a power series representation of π of π₯ in this form by using geometric series. To do this, recall the formula for the infinite sum of a geometric series. If the absolute value of our ratio π is less than one, then the sum from π equals zero to β of π times π to the πth power is equal to π divided by one minus π.

The expression for the infinite sum of our series is very similar to our function π of π₯. One way of seeing this is to rewrite our function π of π₯ as one divided by one minus negative 10π₯. Now, we just need to set our value of π equal to one and our value of π equal to negative 10π₯. This will then give us a power series representation for π of π₯. So by using our formula for the infinite sum of a geometric series with π equal to one and π equal to negative 10π₯, we get that our function π of π₯ is equal to the sum from π equals zero to β of one times negative 10π₯ raised to the power of π. And of course, we can simplify this. Multiplying by one doesnβt change our value.

And by using our laws of exponents, we can distribute the exponent of π over our parentheses. This gives us the sum from π equals zero to β of negative 10 raised to the πth power times π₯ to the πth power. And in fact, we know this has to be a Maclaurin series for our function π of π₯. And the reason for this is, any power series representation of a function π of π₯ centered at a value must be equal to its Taylor series represented at this value.

This means that this series must be our Maclaurin series for π of π₯. Therefore, we were able to show the Maclaurin series for the function π of π₯ is equal to one divided by 10π₯ plus one is the sum from π equals zero to β of negative 10 to the πth power times π₯ to the πth power.

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