Question

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HOW THE SUM OF ALL THE NATURAL NUMBERS IS EQUAL TO -1/12??
==>GIVEN BY श्रीनिवास रामानुजन्

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Answer 1

We start with the series 1-1+1-1+1-1............ Let this series be denoted by S1.
Now - S1 = -1+1-1+1-1+1-1+1........
 Adding 1 to both sides we get 1-s = 1-1+1-1+1-1+1-1.........
                                                       = s
                                                       =>2s =1 => s = 1/2

Now let's look at the series 1-2+3-4+5-6+7-8+9......................
We shall denote it by S2. Now we will add up S2 with itself in this way- 

S2= 1-2+3 -4 +5 -6 +7 -8 +9.........
S2=    1 -2+3 -4  +5 -6 +7 -8

2 S2 = 1-1+1-1+1-1+1-1........... = 1/2
=>S2= 1/4

Now we finally come to the final series S3 i.e. 1+2+3+4+5+...........
-S2 = -1+2 -3+4-5 +6..........
 S3 =  1+2+3+4+5+6..........
=>S3 - S2 = 4+8+12+16+..............................
                 =4(1+2+3+4+5+............)
                 =4S3
=> -S2 = 4S3 - S3 = 3S3
=> S3 = -S2/3
          = (-1/3)(1/4)
          =-1/12 as needed to be shown. The proof may be long, but it's probably the simplest yet.

Answer 2

Hey friend!!!!!!!!
This is my one of the favourite proof .

Let's take three sums
S1=1-1+1-1+1-........
S2=1-2+3-4+5-........
S3=1+2+3+4+5+.........

First,
S1=1-1+1-1+1-........
Now ,The sum fully depend at which point, it will stop.
If it stops at the odd place, then the answer will 1.
If it stops at even place then the answer will be 0.
So,we will take average of these both no. which is 1/2. (1+0/2)


Second,
S2=1-2+3-4+5-........

Now,
2 S2= 1-2+3 -4+5-........
+ 1 - 2+3-4+5-........
______________________
=1-1+1-1+1-........
= 1/2 (S1)
2 S2 =1/2
S2=1/4

Now ,
The last one

S3-S2=(1+2+3+4+5+........)
-(1-2+3-4+5-.............)
______________________
0+4+0+8+0+12+...
=4+8+12+16+.........
= 4 (1+2+3+4+.........)
= 4 S3
Now,
S3-1/4=4 S3. (1/4=S2)
-1/4=3 S3
S3= -1/12
That means (1+2+3+4+.....) or S3 = -1/12

Therefore,The sum of all the natural nos. is -1/12.

PROVED