Does $\sum_{n = 1}^{\infty} \frac{1}{2^n}$ converges?

Question

If I try on a few examples, I see that $\sum_{n = 1}^{\infty} \frac{1}{2^n}$ never goes beyond $1$.

How can I prove it has a limit?

Answer 1

$$\sum_{n=1}^{\infty} \frac{1}{2^n}= \frac {1/2}{1-1/2}=1$$ is a special case of the geometric series $$ a+ar+ar^2+ar^3+... = \frac {a}{1-r}$$ for $|r|<1$

Answer 2

$$ \sum \frac{1}{2^n} = \sum \left( \frac{1}{2} \right)^n = \frac{1}{1-1/2} = 2 $$