Summation (Part 2)

To read Summation Part 1 click here

What is 12+34+56+...1 - 2 + 3 - 4 + 5 - 6 + ... equal?

Let's call the sum ss.

s=(12+34...)s= (1 - 2 + 3 - 4...)

4s=(12+34...)+(12+34...)+(12+34...)+(12+34...)4s= (1 - 2 + 3 - 4...) + (1 - 2 + 3 - 4...) + (1 - 2 + 3 - 4...) + (1 - 2 + 3 - 4...)

4s=(12+34...)+1+(2+34+5...)+1+(2+34+5...)+(12)+(34+56...)4s= (1 - 2 + 3 - 4...) + 1 + (- 2 + 3 - 4 + 5...) + 1 + (- 2 + 3 - 4 + 5...) + (1 - 2) + (3 - 4 + 5 - 6...)

4s=(12+34...)+1+(2+34+5...)+1+(2+34+5...)1+(34+56...)4s= (1 - 2 + 3 - 4...) + 1 + (- 2 + 3 - 4 + 5...) + 1 + (- 2 + 3 - 4 + 5...) - 1 + (3 - 4 + 5 - 6...)

4s=1+(12+34...)+(2+34+5...)+(2+34+5...)+(34+56...)4s= 1 + (1 - 2 + 3 - 4...) + (- 2 + 3 - 4 + 5...) + (-2 + 3 - 4 + 5...) + (3 - 4 + 5 - 6...)

4s=1+[(122+3)+(2+3+34)+(344+5)+(4+5+56)+...]4s= 1 + [(1 - 2 - 2 + 3) + (- 2 + 3 + 3 - 4) + (3 - 4 - 4 + 5) + (- 4 + 5 + 5 - 6) + ...]

4s=1+[0+0+0+0+...]4s= 1 + [0 + 0 + 0 + 0 + ...]

4s=14s= 1

s=14s= \frac {1}{4}!!!

Note by Sharky Kesa
5 years, 10 months ago

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1 vote

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It is also \infty and -\infty if you combine them in a different way.

Bogdan Simeonov - 5 years, 10 months ago

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I've written a post about this sum. This is equal to η(1) \eta(-1) and is factually equal to a quarter, as you said! It is not just a fallacy. See eta function here

Bogdan Simeonov - 5 years, 10 months ago

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It is an intermediary sum for the infamous 1+2+3+4+5+6+=?1+2+3+4+5+6+\ldots=? summation.

Nanayaranaraknas Vahdam - 5 years, 6 months ago

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