Question

What's wrong?
1+2+4+8+16+..... =?
if 2* (1+2+4+8+16+....) -1 (1+2+4=8+16+...) = (1+2+4+8+16+...)
However, if we distribute 2 into terms on the first term, we have 2+4+8+16+.....
so that the first term is 1 less than the second term. And the expression =-1. How??
Please, explain

Answer 1

you're assuming the infinite geometric series converges

Answer 2

This is the tricky thing about infinite series: normal rules of algebra and math in general don't properly apply.

Answer 3

@ganeshie8 I read "How not to be wrong" , this is one of insane problem

Answer 4

however -1 is not a nonsense number :) read below :
http://en.wikipedia.org/wiki/Ramanujan_summation

Answer 5

Yeah!! It's so nonsense, right? sum of a bunch of positive number = a negative one!! ha!!

Answer 6

your work is equivalent to using the infinite *converging* series formula :

\[1+2+4+8 + 16 + \cdots = \dfrac{1}{1-2} = - 1\]

Answer 7

its mostly nonsense, but ramanujan assigned some numbers to these diverging series and analyzed them... they're fun to read.... go through that wiki link :)

Answer 8

Thank you. This book makes me craaaazy

Answer 9

"How not to be wrong" is the book name is it ? looks like an interesting read xD

Answer 10

http://www.amazon.com/How-Not-Be-Wrong-Mathematical/dp/1594205221

Answer 11

will look for that in library thank you

Answer 12

:)

Answer 13

http://books.google.co.in/books?id=2k9KAgAAQBAJ&dq=how+not+to+be+wrong&hl=en&sa=X&ei=gYfeU6TVOsaPuAS3jYDIAw&sqi=2&ved=0CBsQ6wEwAA

Answer 14

Check this link
http://books.google.co.in/books?id=2k9KAgAAQBAJ&dq=how+not+to+be+wrong&hl=en&sa=X&ei=gYfeU6TVOsaPuAS3jYDIAw&sqi=2&ved=0CBsQ6wEwAA

Answer 15

is it given why is it not wrong in the book

Answer 16

hehe the sum of all even numbers converge to negative result .
also there is many approches like this i get to know xD
amazing i guess

Answer 17

\[\large 1-1+1+1-1+\cdots = \dfrac{1}{1-(-1)} = \dfrac{1}{2}\]

Answer 18

wowi read this yesturday O.O
1-1+1-1+1-1+...

Answer 19

\(\large S = 1-1+1-1+\cdots \)

\(\large S-1 = -1+1-1+\cdots \)
\(\large ~~~~~~~~ = -(1-1+1-\cdots) \)
\(\large ~~~~~~~~ = -S \)

\(\large \implies 2S =1 \)

\(\large \implies S =\dfrac{1}{2} \)

Answer 20

but i read something diffrente
it was to convice that
1-1+1-1+1-1+1_.. is diverge

Answer 21

ofcourse it diverges and we can prove it using alternating series test

Answer 22

but all these calculations are for analyzing diverging series only as we don't want to forget them by simply saying they diverge.

Answer 23

ik ^^
i only said what was the topic i read it in xD