Question

Give an intuition as to why 1/2 + 1/4 + 1/8 + 1/16 + ..... = 1

Answer 1

becasue you are slowly adding smaller values

Answer 2

if I start from here and work myself half way each time; i will eventually get close to you, even tho I may never reach you

Answer 3

I don't think that's enough, the series 1/2 + 1/3 + 1/5 ... doesn't add to 1.

Answer 4

lol ... you asked for intuition :)

Answer 5

Yes, but I meant something that would be accurate with the series converging to the value of 1. But I get your point.

Answer 6

Maybe think of a square that is 1 by 1. Now color in 1/2 the area. You have the other 1/2 blank. Now color in 1/2 of the remaining (which is 1/4). Continue to color in half of what is remaining and you'll get closer and closer to coloring in the entire square, which has area = 1.

Answer 7

amistre is absolutely right...
if u see 1/2 is half of one, so u hv covered 1/2 the distance to 1
then 1/2 of 1/2 = 1/4 , so u hv covered 3/4 of total distance to 1
then 1/2 of 1/4 = 1/8 and so on

u will get very very very near to one but never to one theoretically
however for practical purposes u will hv reached 1

Answer 8

joebrown: That's what I was looking for. :) But everybody gets a medal.

Answer 9

oh ... a geometric interpretation ;) my confusion was in the word "intuition" :)

Answer 10

Now go to my other question. Of why 0. 999999.... and 1 are the same.

Answer 11

whatever route one take, this will also lead to the square being "nearly" filled and not completely filled.....

Answer 12

if i take a carrot and chop it in half each time, I will get ever so close to chopping up the whole thng

Answer 13

there are a few reasons why .9999.... equals 1

Answer 14

the obvious is that 1/3 is a real number that decimates to .3333.......

Answer 15

1/3 + 1/3 + 1/3 = 1

.333... + .333... + .333... = .999... = 1

Answer 16

If you let x = 0.9999999...
Then that means that 10x = 9.999999....
Then 10x - x = 9.99999... - 0.9999...
And 9x = 9
So x = 1.

Answer 17

Those are both great ways of proving it. ^^

Answer 18

as they say "A cat can be skinned in more than one way....."

Answer 19

.99999 = 9(.1) + 9(.01) + 9(.001) + ...

Answer 20

\[\frac{ar}{1-r}=1\]
\[\frac{9(1/0)}{1-(1/10)}=1\]

Answer 21

ack .... typing and mathing dont mix

Answer 22

We really had some brain storming here. :D