Thread: Find a power series representation for 2xarctan(8x).

I know that d/dx arctan(x) can be represented by finding its derivative, which is 1/(1+x^2). However, when I take the derivative of 2xarctan(8x), I get 2arctan(8x)+2x/(1+(8x)^2), which isn't in the correct form for a power series. So now I don't know where to go.

Thanks for helping!

Originally Posted by jay0812

I know that d/dx arctan(x) can be represented by finding its derivative, which is 1/(1+x^2). However, when I take the derivative of 2xarctan(8x), I get 2arctan(8x)+2x/(1+(8x)^2), which isn't in the correct form for a power series. So now I don't know where to go.

Thanks for helping!

no, first find the power series for arctan(8x), with the method you described. then multiply through by 2x

Is there a particular reason why I am able to just ignore the 2x until it's convenient to put in? It almost seems too easy.

Originally Posted by jay0812

Is there a particular reason why I am able to just ignore the 2x until it's convenient to put in? It almost seems too easy.

do you have a problem with easy? would you rather it be difficult?

yes, the reason is this. the power series in question converges to arctan(8x), and since 2xarctan(8x) = 2x*arctan(8x), we can simply replace the latter term with its power series, as it is the same thing, and then keep the 2x in front and multiply through. this works because of the convergence of the power series to the function.

Of course I'd rather it be easy; I just like to know why I'm able to use certain tricks. =) Thanks for the explanation!