Question

prove 1+2+3+4+5+6+7+8+9........=-1/12​

Answer 1

Answer:

-1/2 this

answer

Step-by-step explanation:

-3c=1-2+3-4+...=1/(1+1)2

=-1/12

Answer 2

Let A = 1-1+1-1....... -> eq.1

1-A = 1-(1-1+1-1.......)

1-A = 1-1-1+1-1.......

1-A = A [from eq.1]

1 = 2A

A = 1/2 -------> eq.2

Let B = 1-2+3-4+5...... -> eq.3

A-B = (1-1+1-1+1.....) - (1-2+3-4....)

A-B = 1-1+1-1...... -1+2-3+4......

Rearranging,

A-B = (1-1) + (-1+2) + (1-3) + (-1+4).....

A-B = 1-2+3-4.....

A-B = B [from eq.3]

A = 2B

1/2 = 2B [from eq.2]

B = 1/4 --------> eq.4

Let C = 1+2+3+4+5+....... ->eq.5

B-C = (1-2+3-4+5.....) - (1+2+3+4+.....)

B-C = 1-2+3-4.... -1-2-3-4.....

B-C = (1-1) + (-2-2) + (3-3) + (-4-4)......

B-C = -4 -8 -12.....

B-C = -4(1+2+3+4+.....)

B-C = -4C [from eq.5]

B = -3C

1/4 = -3C [from eq.4]

C = -1/12

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

Proved