1+2+3+4+5+.....= -0.083333.. ​1^3+2^3+3^3+4^3+.....=?

1+2+3+4+5+.....= -0.083333.. This is the sum of Riemann's zeta function at s=-1

1^3+2^3+3^3+4^3+.....=ζ(3) = 1.20205690315959428539973..........etc.

This is called "Aprey's constant", which is the sum of Riemann's zeta function at s=3

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The first line is contradictory. How can a bunch of positive integers add up and became a negative fraction which is $-\dfrac{1}{12}$?

But I will solve this.

$1+2+... = -\dfrac{1}{12}\\ 1^3+2^3+...... = (1+2+.... )^2=(-\dfrac{1}{12})^2=\dfrac{1}{144}$

This is still contradictory because the sum of a bunch of third powers of a negative number is a positive number.