1 - 2 + 3 - 4 + 5 - 6 + ... made easy | Hacker News Reader

1 - 2 + 3 - 4 + 5 - 6 + ... made easy

paulbuchheit.blogspot.com

49 pointsvinutheraj17 comments

Open articleView on HN

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

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

Wolfram Alpha continues to impress me.

But not me

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

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+-....

hmmm ... for whatever reasons the link was broken. Trying it again: "http://www.wolframalpha.com/input/?i=1+-+2+%2B+3+-+4+%2B+5+-...

here is a PDF snapshot.

"http://www.wolframalpha.com/input/pdfGet.jsp?id=MSP55619821e...

Although I'm not a fan of using i in a mathematical formula like this. When I first saw it, it looked like they meant sqrt(-1).

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...

Almost all mathematical proofs are reverse engineered.

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.

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.

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.

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/Telescoping_series

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.

If she has nothing better to do at the moment.

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.

Reply on news.ycombinator.com