If-they-said-that-k-1-k-diverges-why-1-2-3-4-1-12-

Question Number 9436 by geovane10math last updated on 08/Dec/16

If they said that \underset{k = 1}{\sum^{\infty}} k diverges, why

1 + 2 + 3 + 4 + \dots = - \frac{1}{12} ?

Answered by mrW last updated on 08/Dec/16

T = 1 - 2 + 3 - 4 + 5 - 6 + \dots

T = 1 - 2 + 3 - 4 + 5 - 6 + \dots

T = 1 - 2 + 3 - 4 + 5 - 6 + \dots

T = 1 - 2 + 3 - 4 + 5 - 6 + \dots

4 T = 1 + 0 + 0 + 0 + 0 + 0 + \dots = 1

T = \frac{1}{4}

1 + 2 + 3 + 4 + 5 + 6 + \dots = S \dots (i)

1 - 2 + 3 - 4 + 5 - 6 + \dots = T \dots (ii)

(i) - (ii) :

2 \times (2 + 4 + 6 + 8 + 10 + \dots) = S - T

4 \times (1 + 2 + 3 + 4 + 5 + \dots) = S - T

4 S = S - T

3 S = - T = - \frac{1}{4}

S = - \frac{1}{12} = 1 + 2 + 3 + 4 + 5 + \dots

Commented by geovane10math last updated on 09/Dec/16

Thanks

Answered by mrW last updated on 09/Dec/16

other strange things :

S = 1 + 2 + 3 + 4 + 5 + 6 + \dots (i)

S = 1 + 2 + 3 + 4 + 5 + \dots (ii)

(i) + (ii) :

2 S = 1 + 2 + 4 + 6 + 8 + 10 + \dots

2 S = 1 + 2 (1 + 2 + 3 + 4 + 5 + \dots)

2 S = 1 + 2 S

0 = 1

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