Question

prove it..

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

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Answer 1

hey there ,,

let us check out the solution ;-

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

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let s = 1+2+3+4+5+6+7

consider , s1 = 1-1+1-1+1-1+1-1...........

now ,

this sum should be 0 or 1

if even than , s1 = 0
but , if odd , s1 = 1

s1 = (1-1+1-1+1-1+1-1.......)

1-s1 = 1-(1-1+1-1+1-1+1-1.......)

1-s1 = s1

so , s1 = 1/2

hmm ,,, for now on , we could agree with s1 = 1/2 .

let , s2 = 1-2+3-4+5-6+7...............

so , s2 = 1-2+3-4+5-6+7-8+-9.........

s2 = 1-2+3-4+5-6+7-8. ( shifted rhs byva unit position )

2s2 = 1-1+1-1+1-1+1......

hence ,

2s2 = s1

Therefore , s = 1/4

let's come to our sum of all natural numbers .

s = 1+2+3+4+5+6+7+8+9.........

s2 = 1-2+3-4+5-6+7-8+9.......

so ,

s-s2 = 4+8+12+16+20


hence ,

s-s2 = 4(1+2+3+4+5+6+7+8.......)

s-s2 = 4s

so ,

-s2 = 3s

And , s = -s2/3 = -1/12

hence ,

sn = -1/12. ( sn = sum of all natural numbers ).

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hope it helps @amrit

be brainly , together we go far ☺♥

Answer 2

The answer is explained in the attachment.

Hope it helps!