>>101355
The above manipulations do not consider what the sum of a series actually means. Still, to the extent that it is important to be able to bracket series at will, and that it is more important to be able to perform arithmetic with them, one can arrive at two conclusions:
The series 1 − 1 + 1 − 1 + … has no sum.
...but its sum should be 1/2.
In fact, both of these statements can be made precise and formally proven, but only using well-defined mathematical concepts that arose in the 19th century. After the late 17th-century introduction of calculus in Europe, but before the advent of modern rigor, the tension between these answers fueled what has been characterized as an "endless" and "violent" dispute between mathematicians
...
In modern mathematics, the sum of an infinite series is defined to be the limit of the sequence of its partial sums, if it exists. The sequence of partial sums of Grandi's series is 1, 0, 1, 0, …, which clearly does not approach any number (although it does have two accumulation points at 0 and 1). Therefore, Grandi's series is divergent.
Но блядь, результаты же используют в моделях теории струн?
http://en.wikipedia.org/wiki/Riemann_zeta_function