Since [tex]\frac{1}{2^{n}}[/tex] is a geometric series, for a series like this to converge, its ratio has to be less than 1.
The ratio is [tex]|\frac{1}{2}| < 1[/tex], thus, we can conclude that [tex]\sum_{n=1}^{\infty} \frac{1}{2^{n}}[/tex] will converge (ie has a limiting sum)
