Euler's 1-2+3-4+5 = 1/4 Formula - What's Your Experience?

[member 141513]

1-2+3-4+5... = 1/4

i know this was told by Euler longlong time ago in 18th century
but may i know what do u really feel in daily experience and also math sense

thank you=]

 

[pwsnafu]

http://en.wikipedia.org/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%C2%B7_%C2%B7_%C2%B7" explains it in detail.

In short, the sum does not converge, so that is not the usual definition of equality. But you can http://en.wikipedia.org/wiki/Divergent_series#Abel_summation" it. We consider

1 - 2 x + 3 x² - ...

and observe that for 0 < x < 1, this converges to (1 + x)^-2, and further that the limit x -> 1 gives the answer 1/4. We then "define" the sum to be this.

Summation methods such as http://en.wikipedia.org/wiki/Ces%C3%A0ro_summation" are not "bad" in any sense, just a different way at looking at sums.

 

[member 141513]

http://en.wikipedia.org/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%C2%B7_%C2%B7_%C2%B7" explains it in detail.

In short, the sum does not converge, so that is not the usual definition of equality. But you can http://en.wikipedia.org/wiki/Divergent_series#Abel_summation" it. We consider

1 - 2 x + 3 x² - ...

and observe that for 0 < x < 1, this converges to (1 + x)^-2, and further that the limit x -> 1 gives the answer 1/4. We then "define" the sum to be this.

Summation methods such as http://en.wikipedia.org/wiki/Ces%C3%A0ro_summation" are not "bad" in any sense, just a different way at looking at sums.

but what is ur feeling about this equation (without using math angle of view)is it stunning or else=]?