Exponential function-like Taylor series: what is it?

Question

I have a series $$1+ x+\frac{x^2}{2}+\frac{x^3}{4}+\frac{x^4}{8}...=1+\sum_{n=1}^\infty \frac{x^n}{2^{n-1}}$$

that looks an awful lot like a Taylor series of some kind. If the denominator of the fraction in the summation were $n!$ instead of $2^{n-1}$ we would have the Taylor series of $e^x$, expanded around $x=0$. What Taylor series is this?

Answer 1

Hint

Can you evaluate

$$\sum_{n=1}^\infty \frac{x^n}{2^{n-1}}=2\sum_{n=1}^\infty \frac{x^n}{2^n}=2\sum_{n=1}^\infty \left(\frac{x}{2}\right)^n?$$

Answer 2

Hint: Divide the entire series by $2$, and then add $\frac 12$. You'll get something nicer.