1Right.Thanks [122]
If S=1+2+4+8+16+...∞.Then S is a positive number.Multiply both sides by 2,then it is found that 2S=S-1 which leads to conclusion that S=-1 which is certainly negative.Do you agree with conclusion?Give a suitable explaination too
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6 Answers
1u cant do
∞=∞
the series is divergent
so the assumption that it converges to a S is absurd
that is
2S=S-1 is absurd
341However: http://en.wikipedia.org/wiki/Divergent_series talks of how you can assign the value of -1 to this series under certain cionditions
341What gr8dreams is saying is this: The laws of cancellation for the addition operation are valid only for the field of real numbers.
2S = S-1 gives S=-1 as a result of the cancellation law.
So, before you use the law, you must be satisfied that S is a real number and ∞ does not belong in the real number system
1yes , actually i was very hesitant to respond to this thread ,
i mean this topic i way beyond jee level .......
also see this ..
http://en.wikipedia.org/wiki/Ramanujan_summation
and also this
http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%C2%B7_%C2%B7_%C2%B7
ramanujan the greatest mathematician of india himself as told
"
The Ramanujan sum of 1 + 2 + 3 + 4 + · · · is also −1â„12 In Srinivasa Ramanujan's second letter to G. H. Hardy, dated 27 February 1913, he wrote:
"Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1â„12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter. …"
"
62yup and it is interesting bcos you point our ramanujan..
What is important here to note is that the difference between any circle and a real line of of just one point! (Think of it like a value from [0, 2pi) to (-infinity to + infinity.)
taking tan (theta/2)
The only point in the first interval is pi where the function is not defined...
if we were to define infinity and take the interval with another point infinity, then we would have a one one onto relation between the two sets..
(Even though I have not said anything about the other similarities that are there, but this relation reveals a lot!)