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hsbhatt

Let S = r - 2r² + 3r³ - 4r⁴+...

Sr = r² - 2r³ + 3r⁴ -

Hence S(1+r) = r - r² + r³ -....

or S(1+r) = r/(r+1)

or S = r/(r+1)²

Now taking limit as r

1, we get

_r

₁ S = 1/4

Also _r1 S = _r1 r - 2r² + 3r³ - 4r⁴+... = 1 - 2 + 3 - 4 + ....

Now 1 - 2 + 3 - 4 + .... = (1+2+3+4+...) - 2(2+4+6+8+...)

= (1+2+3+4+...) - 4(1+2+3+4+...)

= -3(1+2+3+4+...)

Hence 1+2+3+4+... = -1/12.

What's going on here?!!!!!

apurviitjee2008

well this is just an attempt but i think the case goes wrong in the second step where s is multiplied with r especially for the case when r

1
it can also be multiplied with r^2 and in that case a different result is obtained
just an attempt
plz correct me if i m wrong

hsbhatt

But is anyhting wrong with what is done above?

apurviitjee2008

Sir i meant that u will get many solns in that case (inwhich s is multiplied with r and r tends to 1)

it can also be multiplied with r ^2 or r^3 etc

hsbhatt

Hey wait, what did you get by multiplying with r²?

I checked with r³, you get the same result.

It comes back to what is going on here?

apurviitjee2008

Sir it comes something like this

S(1+r²)=r-2r²+3r³-4r⁴+.........

this is an AGP sequence

hsbhatt

Just check your working, I don't think you will be able to evaluate the limit when you mutilpy by r². I mean if you try S(1-r²), you will get a geometric sum, with no limit when r tends to 1.

If you multiply by r³ and in general odd powers of r, you will get the limit as 1/4. So no probs about the value of the limit

apurviitjee2008

yes Sir thats what i m also getting
unable to evalute the limit when multiplied with r^2

apurviitjee2008

Sir then the result does come out to be different for r^2
i think there lies the hint
the limit comes as 1/4 only for odd powers of r

hsbhatt

anyone with any inputs on this one?

sboosy

yes bhattji ..but pls correct me if wrong
actually u have used that
r-r^2+r^3 ...= r/r+1
but sum of infinite series only can b written like that
and for that r<1 and secondly there is some ambiguity
with 1-2+3 ... as it cud b positive for odd n
anyway that i think is not a problem except that r /r+1
i think is not right

hsbhatt

Actually, this is a famous result by a great mathematician. And an accepted result. But it made no sense to me. Apparently it does to a lot of guys. I wanted to see your reactions to it.

Just check: http://www.mathlinks.ro/viewtopic.php?t=78489

sboosy i compared the limit of two equivalent expressions when r tends to 1. I didnt equate them for r=1.

neeraj_agarwal_1990

*deleted*

sboosy

My point is sir ..how in the first place u got r-r^2+r^3...as r/r+1

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