Infinite Geometric Progression

$\displaystyle \begin{aligned} S & = 1 + \frac 12 + \frac 14 + \frac 18 + \cdots \\ & = 1 + \frac 12 \left(1 + \frac 12 + \frac 14 + \cdots\right) \\ & = 1 + \frac 12 S \\ \implies \frac 12S & = 1 \\ \implies S & = 2 \end{aligned}$

$\large \sum _{ n=0 }^{ \infty }{ \frac { 1 }{ { 2 }^{ n } } =\frac { 1 }{ 1-\frac { 1 }{ 2 } } =\frac { 1 }{ \frac { 1 }{ 2 } } =\frac { 2 }{ 1 } =\frac { a }{ b } }$

So, a + b = 2 + 1 = 3.

We can see that a/b is the sum of an infinite Geometric Progression, in which all the terms are multiples of a number a, and here a=1. Also, here there is a constant r, and r is increasing exponentially. Here r=1/2. Thus, r<1. Then, using the formula S= a/1-r, we get S=1/1-1/2. Thus S=1/1/2=2/1. Therefore a/b =2/1. Hence a+b =2+1=3.