Haunted sum

Calculus Level 4

What can you say about the covergence of the following sum?

A = 1 − 2 + 3 − 4 + . . . A=1-2+3-4+ ...

We cannot say anything about it.. It diverges to both + ∞ +\infty and − ∞ -\infty It diverges to − ∞ -\infty It diverges to + ∞ +\infty

Snehal Shekatkar by Snehal Shekatkar , 32, India

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2 solutions

Mirza Baig
Feb 20, 2014

We can say something: It doesn't converges.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

I thought it was 1/4? Numberphile

Spock Weakhypercharge - 7 years, 3 months ago

yeah .that should have been one of the options.

ibrahim abdullah - 7 years, 3 months ago
Benjamin Wong
Feb 19, 2014

its sum oscillates between positive unity and negative unity, so we cannot say anything about it

1 Helpful 0 Interesting 0 Brilliant 0 Confused

There is actually proof that this particular, oscillating sum does have a result, which is quite surprising. So long as we take it to be an infinite sum the result is actually 1/4. It may sound strange (and I'm quite disappointed the author of this question was unaware) but there is proof, which can then be used to prove that the infinite sum 1 + 2 + 3 + 4.... actually equals -1/12 (a result heavily used in physics, particularly string theory).

Have a look on the Numberphile website (top result when you Google Numberphile) and it should be easy to find the video with the proofs. Also, if you're interested in finding out new things about mathematics, browse the rest of the videos.

Alex Panebianco - 7 years, 3 months ago

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Dear Alex, you don't have to get disappointed at all because author is perfectly aware of 1/4 issue.. ;-)

Its not true that this series sums to 1/4, rather, it is a generalized sum of this series which is equal to 1/4 for a particular "analytical continuation" of this series. If you consider different analytic continuation then this series may "sum to" something else. (Consider for example Cesaro summation which doesn't give 1/4). When you talk about normal sum, it doesn't have any value as its sum and we cannot say anything about it. I have never read "string theory" which you may have, but I am sure that even there sum simply means sum of the analytically continuation and not the normal sum.

Snehal Shekatkar - 7 years, 3 months ago

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Ahh, in that case I apologise. I had a quick look at Cesaro summation (and the definition of analytical continuation) and realise your point. I believe the result used for the infinite sum, which plays a part in string theory, may indeed make use of analytical continuation.

Thank you for filling in my blanks. :)

Alex Panebianco - 7 years, 3 months ago

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@Alex Panebianco – Never mind.. :)

Snehal Shekatkar - 7 years, 3 months ago

We can say something: It doesn't converges.

Mirza Baig - 7 years, 3 months ago

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