I have a question about geometric series:

The task wants you to find a function $S(x)$ for the sum of a geometric sequence: $$2\sum_{n=1}^\infty (x-1)^{2n-1} $$

My first thought is to use: $$\sum_{n=0}^\infty r^n*a_0 = \frac{a_0}{1-r}$$

Then rewrite $$2*(x-1)^{2n-1}$$ into: $$2*((x-1)^{2})^{n}*(x-1)^{-1}$$

so $$a_0 = 2*\frac{1}{x-1}$$ and $$r=(x-1)^2$$

I then get:

$$S(x) = \frac{\frac{2}{x-1}}{1-(x-1)^2}$$

which is equal to:

$$\frac{2}{-x^3+3x^2-2x}$$

and the answer should have been:

$$S(x) = \frac{2(x-1)}{1-(x-1)^2}$$

pls help!

## Best Answer

When you use $$\sum_{n=0}^\infty a_0*r^n = \frac{a_0}{1-r}$$ you will have to change the first expression $$2*\sum_{n=1}^\infty (x-1)^{2n-1}$$ into $$2*\sum_{n=0}^\infty (x-1)^{2n+1}$$