# How do you find the Maclaurin Series for #f(x)= x / (1-x^4)#?

I (well, Wolfram Alpha) got:

A Taylor series is written as:

where:

Since you will have to take many derivatives, let's just do them right now. We can try three of them, but even that's a lot, as there will be a lot of quotient rule and chain rule!

And I'll skip the work for the second and third derivatives because you shouldn't need to do this on a test. I just did this in Wolfram Alpha in this case.

So, naturally, we needed to do more derivatives... Great. Anyways, you would (eventually) get:

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To find the Maclaurin series for the function ( f(x) = \frac{x}{1-x^4} ), we first express the function as a geometric series.

[ f(x) = x(1 + x^4 + (x^4)^2 + (x^4)^3 + \ldots) ]

Now, we recognize that this is a geometric series with first term ( a = x ) and common ratio ( r = x^4 ).

The sum of an infinite geometric series is given by:

[ S = \frac{a}{1 - r} ]

Substituting the values for ( a ) and ( r ), we get:

[ f(x) = \frac{x}{1 - x^4} = x(1 + x^4 + (x^4)^2 + (x^4)^3 + \ldots) ]

[ = x + x^5 + x^9 + x^{13} + \ldots ]

This is the Maclaurin series for ( f(x) = \frac{x}{1 - x^4} ).

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