

1 - 2 + 3 - 4 + 5 - 6 + ... made easy - vinutheraj
http://paulbuchheit.blogspot.com/2007/04/1-2-3-4-5-6-made-easy.html

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antimora
Wolfram Alpha has made even simpler, and (more correct proof)

[http://www.wolframalpha.com/input/?i=1+-+2+%2B+3+-+4+%2B+5+-...](http://www.wolframalpha.com/input/?i=1+-+2+%2B+3+-+4+%2B+5+-6+%2B+..).

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tectonic
Wolfram Alpha continues to impress me.

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hyperbovine
But not me

[http://www.wolframalpha.com/input/?i=1+-+2+%2B+3+-+4+%2B+5+-...](http://www.wolframalpha.com/input/?i=1+-+2+%2B+3+-+4+%2B+5+-6+%2B+7..).
.

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vinutheraj
Adding a '+' to the end, before the '..' makes it work again, maybe it needs
an operand and the '..' to show that it is an infinite series, maybe !

[http://www.wolframalpha.com/input/?i=1+-+2+%2B+3+-+4+%2B+5+-...](http://www.wolframalpha.com/input/?i=1+-+2+%2B+3+-+4+%2B+5+-6+%2B+7%2B.).

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tsally
I alway prefer this type of approach for solving problems. Even when formal
proofs are required, I usually get the answer in this way and then reverse
engineer the proof.

Another example (the sum of 1/2 + 1/4 + 1/8 + 1/16...):
[http://media-2.web.britannica.com/eb-
media/79/26979-004-CF3F...](http://media-2.web.britannica.com/eb-
media/79/26979-004-CF3F4DA2.gif)

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eru
Almost all mathematical proofs are reverse engineered.

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spicyj
I find it more helpful to note that

    
    
        1/(1 + x)^2 = 1 - 2x + 3x^2 - 4x^3 + ...
    

for |x| < 1 and then to extend the summation to other values of x.

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mitko
All such attempts for summation are ill-defined. The Abel summation is
actually not a property - it is a _functional_. That said you take the
function f(n)=<n-th summand>. Then you can define AS(f), but it shares very
few properties with normal summation. If you are not careful you can easily
reach contradiction.

For example

1/4 = _1 -2_ +3 _-4 +5_ -6 _+7_ ... = //comment: 1-2-4+5=0,

3-6+9-12... =

3\sum i(-1)^i= 3* 1/4

Contradiction!

Edit: The original contradiction example was more complicated.

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BearOfNH
It's a neat example, but surely you know you can't rearrange terms of an
infinite sum unless it is absolutely convergent. That's not the case here.

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scythe
I personally found the part of the article about Abel summation[1] much more
enlightening. It really seems to indicate _why_ 1 - 2 + 3 ... = 1/4 instead of
just showing that 1/4 arises if you look at the sums a certain way.

[1]:
[http://en.wikipedia.org/wiki/1_−_2_%2B_3_−_4_%2B_·_·_·#Abel_...](http://en.wikipedia.org/wiki/1_−_2_%2B_3_−_4_%2B_·_·_·#Abel_summation)

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jaydub
<http://en.wikipedia.org/wiki/Telescoping_series>

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edw519
_I don't know if a real mathematician would approve of my method, but at least
it's easy to understand._

The good mathematician may not approve.

The better mathematician probably would approve.

The best mathematician would seek and codify a principal to describe the
general case of the instance you cite.

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eru
If she has nothing better to do at the moment.

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thras
It diverges under the ordinary definitions of infinite series, but one can
generalize the notion of summation a little bit to allow for certain divergent
series to have finite sums.

