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Question

1+2+3+4+..... Infinity=? Are in A.P.


Solution

Obviously, it is -1/12. That's definitely the only answer that makes sense (sarcasm intended).

To see why, we should prove it in three parts.
First, consider the series K = 1-1+1-1+1-1....
K+K = 1, so K = 1/2

Second, consider the series J = 1-2+3-4+5-6....
J+J = K, so J = 1/4

Third, consider S = 1+2+3+4+5+6....
S-4S = 1-2+3-4+5-6... = J
so S = -J/3 = -1/12

 

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