What is the Maclaurin Series for f(x)=x/(e^x-1) up to x^2?

[DjDriftX]

Find the first few terms of the Taylor Series around x=0...?
of the function

f(x)= {x/(e^x - 1) , x =/ 0}
{1 , x=0}

the function is piecewise.
up to and including the term involving x^2

It says to not compute derivatives of f but to use the formula for the Taylor series of e^x


x/(e^x-1) = x (1/(e^x-1)
so.
would that be x (1/ \sum(xⁿ/n!) - 1)
or maybe x \sum (1 / (xⁿ/n!) - 1))

I'm not really sure where to start

 

[Dick]

Remember you only need terms up to and including x^2. 1-e^x=x+x^2/2!+x^3/3!+... Divide numerator and denominator by x. Now you've got 1/(1+x/2!+x^2/3!+...). You know an expansion for 1/(1+a), right? Use it. Throw away terms that are higher power than you are looking for.