Maths Best Question

Question

1+2+3+4+5+........Infinite=?

Answer 1

Hey! We can surely say that the sum will be infinity.

But if we go on fun experiment then we will get the sum as -1/12. How?

Let's take sum be S

S=1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + …= ?

S₁ = 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + …

Take a negitive S₁

-S₁ = − 1 + 1 − 1 + 1 − 1 + 1 − 1 + …

Now 1 - S₁ = 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + …

So , 1 - S₁ = S₁

Hence S₁ = 1/2

S₂ =1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + …

S₂ + S₂

=1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + …

+ 0+ 1 − 2 + 3 − 4 + 5 − 6 + 7 + …

=1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + …

= S₁

So,

2S₂ = S₁

2S₂ = 1/2

S₂ = 1/4

S − S₂

=1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + …

− 1 + 2 − 3 + 4 − 5 + 6 − 7 + 8 + …

=0 + 4 + 0 + 8 + 0 + 12 + 0 + 16 + …

= 4( 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + … )

= 4S

So , S - S₂ = 4S

Finally , S – 1/4 = 4S

⇒ S = – 1/12
 Which means

S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + …= – 1/12

Hence proved. But this answer is absolutely wrong as it breaks the sum law.