Understanding Infinity

What is the value of 

 1+ 1/2 + 1/4 + 1/8+ 1/16+ 1/32+ 1/64 +................ ∞  ?

Is it = 2 or ≈(approximately) 2 ? I always thought it to be approximately equal to 2 

Why ? Cause I assumed it to be something like 1.999999999...something.... which was being approximated to 2 due to paucity of space !!

So with this flawed understanding I ventured into calculus and unsurprisingly failed to grasp its essence ,obviously i wasn't WATCHING CLOSELY.

So coming back to our question.To answer that we need to delve into something known as Zeno's Paradox .

Consider a race b/w a hare and a tortoise, where tortoise is given a head-start of 1m (Tortoise starts race 1m ahead of hare)also assume that hare can run with twice the speed of tortoise (let's say their speeds are 2 m/s and 1 m/s respectively.

Zeno's paradox says, given these conditions, hare can never overtake tortoise despite the fact that former runs faster than the latter. Zeno asserts,since the tortoise is given 1m head-start so in order to cross it ,hare would first have to reach the 1m mark & by the time it reaches there, tortoise would have moved by 1/2m (since tortoise's speed is half that of hare's). Now by the time hare catches it at its new position, tortoise would have further by 1/4m more, so each time hare tries to catch it, the tortoise moves to a new position with distance b/w them continuously reducing but tortoise remaining marginally ahead with each iteration !

But a simple algebraic exercise tells us that hare would overtake the tortoise at 2m mark after 1second from the start of the race ! (distance by hare in 1second: 2m/s *1 =2m and by tortoise 1m/s*1=1m and with 1m head-start its total distance too 2m)

So who is correct? On the face of it there seems to be nothing wrong with Zeno's articulation, after-all , in a race you have to first reach your opponent's position in order to beat her and if you aren't able to catch her,question of beating doesn't arise.

Answer to this conundrum lies in our understanding of infinity, since distance covered by tortoise is  1/2 + 1/4 + 1/8+ 1/16+ 1/32+ 1/64 +................ ∞  and by hare 1+ 1/2 + 1/4 + 1/8+ 1/16+ 1/32+ 1/64 +................ ∞  Now if we proceed with our naive understanding of infinity (i.e infinity being some VERY VERY LARGE or VERY VERY SMALL NUMBER etc.) we will never be able to solve this.

So what exactly is infinity then ? Well nobody knows :P ,but humans have found out a way to work with it w/o actually defining it !!

By using various previously proved results and axioms and using them as STEPPING STONES to venture into the unknown.Just like the above case (i.e with known results in real world and algebraic verification of it) we deduced that value of this expression 1+ 1/2 + 1/4 + 1/8+ 1/16+ 1/32+ 1/64 +................ ∞ HAS BE TO EXACTLY EQUAL TO 2 ..Sum of an INFINITE SERIES can be a finite one! 

The purpose is to "TACKLE INFINITY NOT DEFINING IT" coz it is a different beast altogether which is yet to be tamed!