I've looked over this peculiar problem and the many different forums that have tried to resolve it yet i'm confused as to why the answer simply isn't zero. for those not familiar with this problem, it is this(adsbygoogle = window.adsbygoogle || []).push({});

Ʃ of (-1)^n as n goes from 0=∞ which produces a series that equals .......

1+1-1+1-1+1-1...... and so on.

now i have seen some infinite analysis on this equation ( mainly using analysis of continuous functions) and i've seen how people manipulate the commutative properties of addition and subtraction to come up with various answers, but i propose a different way to look at it to be scrutinized:

#1 - each additive element of the equation is based off of integers, and divided quite neatly. if n is an even number then that element of the series is +1, if n is a negative number then that number is -1 so it should conclude that -

#2 - the Grandi's series will end in a number other than zero if there are either more positive numbers or negative numbers in the select group of integers from zero to infinity which n represents which can be shown -

#3 - through a little algebra magic and a couple of simple functions that there are exactly as many even and odd numbers regardless of where you start counting them from as follows -

let x = 0 → ∞

evaluate the equations y = 2x and z = 2x + 1

as you evaluate each equation at the same time it produces evens and odds at the same rate hence establishing a one to one ratio between them which means that the Grandi's series should also have a one to one ratio of +1's and -1's and the answer should be zero.

even if n were to start a 5 → ∞ the same results would show.

i would enjoy being ripped apart by the experts if my logic is wrong......

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# The Sum of the Grandi's Series

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