Puzzles - What's the mistake?

13

Ð”Ð²Ò¥Ñ—ÑuÏ now in medical c ·2009-03-14 04:56:40

@ Chetan

the sum is infinite for the second one..

and for the first one it is either 0 or +1...not +1 or -1

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33

Abhishek Priyam ·2009-03-16 02:28:18

1 + 1 + 1 + 1 + Â· Â· Â·, also written , is a divergent series, meaning that it does not have a sum in the usual sense. Its partial sums increase without bound.

Where it occurs in physical applications, 1 + 1 + 1 + 1 + Â· Â· Â· may sometimes be interpreted by zeta function regularization. It is the value at s = 0 of the Riemann zeta function

in which the function is defined where the series diverges by analytic continuation. In this sense 1 + 1 + 1 + 1 + Â· Â· Â· = Î¶(0) = âˆ’1â„2.

___________________________________

Means in physical applications it is taken as -1/2 [5][11]

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341

Hari Shankar ·2009-03-16 02:21:57

I stumbled across these summations sometime last year. There were many mathematicians who wondered whether at all they made sense. But Leibnitz said that it appealed to his intuition that nature when confronted with a choice (like in 1-1+1... we have 0 and 1 as possible sums) nature would choose their average.

And this finds an application in quantum physics where they often deal with divergent sums this way and the theories work!

By the way if you look up Abel Summation on wikipedia you will get some theorems on summing up of divergent series!!!!

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62

Lokesh Verma ·2009-03-16 02:15:10

thanks prophet for these links

I did not know about casero summation..

did hear about ramanujan summation though

This is one quote from the links givenby you

" I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + Â· Â· Â· = âˆ’1â„12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal." - Ramanujan :)

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341

Hari Shankar ·2009-03-16 01:22:14

http://en.wikipedia.org/wiki/CesÃ ro_summation maybe relevant

A greater surprise awaits us at http://en.wikipedia.org/wiki/Ramanujan_summation

By now http://en.wikipedia.org/wiki/1_%2B_1_%2B_1_%2B_1_%2B_Â·_Â·_Â· will not be a surprise :D

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1

Akand ·2009-03-16 00:48:56

i thought tht was obvious...........S=0......

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66

kaymant ·2009-03-16 00:46:26

The trouble is that the object on the right is simply not there so one cannot assign it to S and then pretend that S behaves like a normal mathematical object.

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62

Lokesh Verma ·2009-03-16 00:14:00

Nitin you are right.. but the thing is that here the flaw is not because of that!

The flaw in the above proof has nothing to do with r<1

It is basically a condtion for convergence, which you will study in a real mathematics course. The understanding of which is very much possible at this stage :)

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1

nitin_saini ·2009-03-16 00:08:29

infinite sum z possible if r<1 or r >-1...in this case answer z infinity

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62

Lokesh Verma ·2009-03-15 23:51:17

I wud put it this way

our basic assumption that

1-1+1-1+1........ =S

is not correct

basically this sum does not exist.. so where is the question of assuming that this sum is equal to something!

Hence the fallacy..

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13

Ð”Ð²Ò¥Ñ—ÑuÏ now in medical c ·2009-03-14 05:27:25

yup...gordo i told the same in post#12

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1

gordo ·2009-03-14 05:25:14

the thing is...the one that u said...ie S=1+2+4+8+...
or S= 1+ 2(1+2+4+..) or S=1+2S...
is OK till there...but further attempts to solve the qn makes no sense...becoz of the simple reason that by common sense we know that S is infinity...a term that no one knows..a value that no one can interpret...a value that has no place in a mathematical eqn. where we wanna get some interpretation...so in this eqn. we have 2 such terms...to a further attempt to solve the eqn. wud be senseless...
its like telling 4infinity =2+ 2infinity...which makes no sense at all...

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13

Ð”Ð²Ò¥Ñ—ÑuÏ now in medical c ·2009-03-14 00:02:36

bhaaiya....

actually we don't know ...whether infinity is even or odd...

so the sum itself has two probable values...1 or zero

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1

chetan wahi ·2009-03-14 01:43:38

@abhirup... S is not infinity.

is it? S is only 1 or -1 as you can see.
S can never become infinite.

So that logic will not work here..

But I am sure this is something wrong somewhere..
Nishant Bhaiya had given the mistake in the
S=1+2(1+2+4+8+....) question long back if i remember correctly. But Is that applicable here..