Question

1+2+3+4+5+6+7+8+9+...................................................... = -1/12.........
prove if you are not a fake genius ..................♥♥♥

Answer 1

ANSWER
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1 + 2 + 3 + 4 + 5 + 6 + .......... = -1/12

By Ramanujan sum of Infinity


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Sorry I am not a genius but Ramanujan was.

^_^

Answer 2

$\huge\boxed{Answer}$

1 + 2 + 3 + 4 + 5 + 6 + ..................... = -1/12

Let , There are n terms in this sequence ,

Then An = a + (n - 1)d

= 1 + (n - 1)(1)

= 1 + n - 1

= n

Given , Sn = -1/12

n/2 {a + An} = -1/12

n/2 {n + 1} = -1/12

n (n + 1) = -1/6

6n(n + 1) = -1

6n² + 6n + 1 = 0

6n² + 6n + 3/2 + (1 - 3/2) = 0

(6n² + 6n + 3/2) - (1/2) = 0

3(2n² + 2n + 1/2) - (1/2) = 0

(2n² + 2n + 1/2) - (1/6) = 0

(4n² + 4n + 1)/2 = 1/6

(4n² + 4n + 1) = 1/3

3

(2n + 1)² = 1/3

2n + 1 = 1/√3

2n + 1 = 1/√3

2n = (1 - √3)/√3

n = (1 - √3)/2√3

Proof :-

Sn = (1 - √3)/4√3 { 1 + (1 - √3)/2√3) }

= (1 - √3)/4√3 + (1 - √3)²/24

= {(1 - √3)(2√3) + (1 - √3)²} / 24

= { 2√3 - 6 + 1 - 2√3 + 3 } / 24

= (- 6 + 4)/24

= -2/24

= -1/12

I Hope it will be helpful for u