Thread: series representation

Represent the function f(x) = x/(1+4x)^2 as a power series and determine the radius of convergence.

so the correct answer is (-1)^n * (n+1) *4^n*x^(n+1) but this is what I got:

x/(1+4x)^2

x/[1-(-4x)^2

x series sign (-4x)^n

series sign (-1)^n * 4^n x^(n+1)

Where did the (n+1) in front of the 4^n come from?

Sorry, but I can't understand what you mean. What is the question exactly?

so the correct is what I have multiplied by (n+1)

And what does this mean? What have you multiplied by (n+1) and where?

Are you starting with the power series



and if so, what are you doing to this series to result in your series?

OK so I fixed the question. What I get from this section is that you want to make your function look like 1/1-x and go from there according to the definition.

Right, so differentiate my series



so



or



or




Now replace with and multiply the whole thing by .

So once you have the power series representation of a function, you the take the derivative?

Originally Posted by smray7

So once you have the power series representation of a function, you the take the derivative?

You showed in the OP that you already know the series for 1/(1 - x). The derivative of that series is the series of that derivative, -1/(1 - x)^2. This more closely resembles the original problem which you are working towards.

This is just one tool you can use to find a series for a function you didn't know.

You could show that, if , then:



and use the general formula for Taylor/Maclaurin series.