Question

what is the sum of all naturql number upto infinity

Answer 1

$Hey \: \: There \: \:$
I'm gonna give you the solution which is nothing but a shadow of the ramanujan's proof

we know

the sum of n terms is given by

$Sn \: = \frac{n \: }{2} (2a \: + (n - 1)d)$
let us coming to our Question ,
you know what the sum of all natural numbers that's [ 1 + 2 + 3 + 4 ...... -1/12 ] which is also used in the string theory

Proof :- S = 1 +2+3+4+..... = -1/12


Let's consider

$S \: \: = 1 + 2 + 3 + 4 + 5 + 6..... \infty \\ \\ now \: multiply \: it \: by \: 4 \\ \\ 4S \: = 0 + 4 + 0 + 80 + 12 + 0 + 16 + ...... \\ \\ subtracting \: this \: by \: - 3 \\ \\ - 3S \: = 1 - 2 + 3 - 4 + 5 - 6 + .... \\ \\ further \: \\ \\ \frac{1}{ {(1 + x)}^{2} } = 1 - 2x + 3 {x}^{2} - 4 {x}^{3} + ..... \\ \\ putting \: x = 1 \: we \: get \: \\ \\ - 3S \: = 1 - 2 + 3 - 4 + 5 - 6 + ...... \\ \\ = \frac{1}{(1 + {x})^{2} } = \frac{1}{4} \\ \\ then \: dividing \: by \: - 3 \\ \\ S = \frac{ - 1}{12}$


Hope this helps ya ☺