pwsnafu said: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.
Euler's 1-2+3-4+5 = 1/4 formula is a mathematical formula that was discovered by the famous mathematician Leonhard Euler in the 18th century. It is also known as the alternating harmonic series and it states that the sum of the series 1-2+3-4+5... to infinity is equal to 1/4.
The formula works by using the concept of telescoping series, where each term in the series cancels out with the next term, leaving only a finite number of terms. In this case, the terms alternate between positive and negative integers, and the sum of these terms approaches 1/4 as the series goes to infinity.
Euler's 1-2+3-4+5 = 1/4 formula has many applications in mathematics and physics. It is linked to the Zeta function, which has important implications in number theory. It is also used in the study of divergent series and has connections to the Riemann hypothesis. In physics, it has been used to calculate the energy of the quantum vacuum.
Yes, there is some controversy surrounding Euler's 1-2+3-4+5 = 1/4 formula. Some mathematicians argue that it is not a valid mathematical result because the series does not converge in the traditional sense. However, others argue that it is a useful tool in certain contexts and can be justified through rigorous mathematical reasoning.
There are several ways to prove Euler's 1-2+3-4+5 = 1/4 formula, depending on the mathematical approach used. One way is to use the Riemann zeta function and complex analysis. Another way is to use analytic continuation, a technique used to extend the domain of a function beyond its initial definition. Some other methods include using regularization techniques and Abel summation.