Answer
The series converges. The sum of the infinite geometric series is $1 + \sqrt 2$
Work Step by Step
The series is convergent because |r| $\lt$ 1.
r (the common ratio) = $\frac{1}{\sqrt 2}$ since each term is multiplied by $\frac{1}{\sqrt 2}$.
Use the formula for the sum of an infinite geometric series: S = $\frac{a}{1-r}$ and plug in the first term, $\frac{1}{\sqrt 2}$, for a, and the common ratio, $\frac{1}{\sqrt 2}$ for r: S = $\frac{\frac{1}{\sqrt 2}}{1 - \frac{1}{\sqrt 2}}$ = $1+\sqrt 2$