Find the function that gives the following Taylor series centered at 0

Question

I have to solve this kind of exercise where I have to find a function $f:]-1,1[\rightarrow\mathbb{R}$ that have the following Taylor series centered at $0$

$$ 2+2x+2x^2+2x^3+2x^4+2x^5+... $$

My teacher said, that i have to imagine the Taylor series as a series from calculus 1. I have to check what that series converges against and I also have to check if the function gives the above Taylor series.

I haven't worked with this kind of exercise before, and i hope i can get some help.

Thanks in advance.

Answer 1

I suppose you could just:

$$2[1+x+x^2+x^3+......]$$

The term in square brackets is an infinite geometric progression

So,

$$=2\frac{1}{1-x} \quad \forall \ |x|<1 (\text{which is given})$$

Thus your function must be $f(x) = \frac{2}{1-x}$