MHB Nick's question at Yahoo Answers (Maclaurin series)

[Fernando Revilla]

Here is the question:

for f(x)= (x)/(1-x^4)

= summation from 0 to infinity:

Here is a link to the question:

Find the Maclaurin series? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Hello nick,

If $|t|<1$ we know that $\dfrac{1}{1-t}=\displaystyle\sum_{n=0}^{\infty}t^n$. Then, using the Algebra of series: $$f(x)=x\cdot\dfrac{1}{1-x^4}=x\sum_{n=0}^{\infty}(x^4)^n=\sum_{n=0}^{ \infty}x^{4n+1}\quad (|x|<1)$$ So, necessarily the Maclaurin series for $f(x)$ is $\displaystyle\sum_{n=0}^{\infty}x^{4n+1}.$