MHB How do I find the MacLaurin series for $\frac{1}{1 - 2x}$?

[tmt1]

I need to find the maclaurin series of the function

$$\frac{1}{1 - 2x}$$.

I know $\frac{1}{1 - x}$ is $1 + x + x^2 + x^3 ...$ but how can I use this to solve the problem? I don't think I can just plug in $2x$ can I?

 

[Greg]

Yes; plug in $2x$ and note that the series converges for $|x|<\dfrac12$.

 

[tmt1]

Yes; plug in $2x$ and note that the series converges for $|x|<\dfrac12$.

So, I would get something like

$$\sum_{n = 0}^{\infty} 2^nx^n$$

Is this correct?

 

[Greg]

Yes, but I think the notation

$$\sum_{n=0}^\infty(2x)^n$$

would be preferred. :)

 

[Prove It]

Yes, but I think the notation

$$\sum_{n=0}^\infty(2x)^n$$

would be preferred. :)

Actually, as it's specifically asked to be written as a MacLaurin Series, where it should be written $\displaystyle \begin{align*} \sum_{n = 0}^{\infty}{c_n\,x^n} \end{align*}$ the OP's notation would be preferred.