An infinite series...

Hi People...
just a few minutes back.. I was onto a discussion with some friends on a proof of '2 = 1'

The proof went -

Let: X=Y

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x+x-x = y+y-y

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2x-x = 2y-y

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2x-2y = x-y

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2(x-y) = x-y

By dividing x-y both side:

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2=1

Now I know a majority will say that there has to be something wrong but what?? It looks perfect...
But unfortunately.. who wrote this was a great doctor but had to learn a few things about math :P

see.. I know its just for fun... what's the harm, however, to learn a new thing??

you say...

x + x - x = y + y - y... See More

which simply is

x = y

which means

x - y = 0

Let x - y = c (a new variable)

then

c = 0

so further you say

2 c = c

now the rule says if there is a number = 0 on both the sides of a mathematical equation, then it cant be cancelled.. as multiplication and division is allowed only with a non zero number...

so the real solution to this question.. will be...

2c - c = 0

c = 0

which is true as proved above!! :P


so you see.. with this proof.. you end up into a loop which will roam around 'c = 0'
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Now try this people...

Ps - you need to know a few terms to get the catch of it.. like infinite and series... these words sound simple but unfortunately arent!! If you dont know them... dont go into detail... just deal with simple math and i have written it in the simplest math I could..

let s be an infinite series... (that is that it goes on and on and on.. no ending)

s = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 ...

s = (2 - 1) - 1/2 + (2-1)/3 - 1/4 + (2-1)/5 - 1/6 + (2-1)/7 ...

s = 2 - 1 - 1/2 + 2/3 - 1/3 - 1/4 + 2/5 - 1/5 - 1/6 + 2/7 - 1/7 ...

rearranging...

s = 2 - 1 + 2/3 - 1/2 + 2/5 - 1/3 + 2/7 ...

taking 2 common...

s = 2 ( 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 ...)

= 2.s


Here I just prooved 's = 2s'

This proof is very well Mathematical.. true and well defined.. there is no flaw...

from here... if someone wants you can conclude 2 = 1... however this is not correct.. there is a flaw and lets see if you find it :D

All I wanna say is... 2 years have been the 'mathematical' years of my life... I have known math more than I knew what is happening to me... Math is a part of my life.. and I learnt it from someone who knew Math more than Math knows itself :P and the other teacher made me do it so much that I can even do it in sleep :P
I love them both for this!! I said so.. because with Math... dont take panga if am around :P