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The Sum of the Grandi's Series

  1. Jun 26, 2013 #1
    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

    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......
  2. jcsd
  3. Jun 26, 2013 #2


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    It diverges as an ordinary series.
    The Cesàro summation is 1/2.
    Other summations can vary.
    You need to specify the exact summation you want.
  4. Jun 26, 2013 #3
    i understand the cesaro summation, but that's a technique used in convergent series, i was trying to give a reasonable explanation as to why the series equaled zero through noncontroversial methods. my thought process here was to give a very strong argument to the summation being zero. while infinite divergent series may also be cesaro summable, i was trying to provide a good example as to why the cesaro sum didn't fit here.
    Last edited: Jun 26, 2013
  5. Jun 26, 2013 #4


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    Grandi's Series is a famous divergent (as an ordinary series). To assign it a value you need to define the exact way you are giving value to divergent series. There are convincing arguments for 0,1/2,1, and other values. Numberphile just did this
    Last edited by a moderator: Sep 25, 2014
  6. Jun 26, 2013 #5
    To put it simply, there is no one good answer, 0,.5 and 1 are all considered to be correct. Check out the Numberphile video that was provided, it's not very mathematically rigorous but it provides a good explanation. Also, remember that this series is a divergent one.
  7. Jun 27, 2013 #6
    i guess what i was trying to do was infuse a bit of set theory into the problem. and in set theory the amount of even integers are equal to the amount of odd integers and you can prove this pretty simply with a couple of equations and demonstrating the one to one corresponding elements in them. my example was to show that there exactly the same countably infinite amount of (+1) elements in the Grandi's series as there are (-1) elements so my argument here is for the answer to be zero. i see the Grandi's series as the exact same problem of 'what is the sum of all integers?'. it should be zero because there are exactly the same infinite amount of negative integers as there are positive integers and they can be paired up infinitely. the same pairing examples for alternate answers (like in the Grandi's series) could be used in this instance where you could group things like - ( 0 + (-2)) + (2 + (-4)) + (4 + (-6))...... which would give you an infinite sum but disguise cleverly that each element has a proven corresponding opposite element that will cancel it out to zero eventually over the course of infinity regardless of how you group or pair them.

    using this thinking and going backwards to say that the grandi's series = 1 is equivalent to saying that there is exactly one more even number than there are odd numbers.
    ty for the numberphile link.
  8. Jun 27, 2013 #7
    and i did make a mistake in my very first posting :

    "#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 -"

    should read -

    #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 an odd number then that number is -1 so it should conclude that -
  9. Jun 27, 2013 #8


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    In set theory there are the same number of evens as odd, but also the same number of primes, algebraic numbers, and rational numbers. That definition is not very useful for series. In some areas like number theory densities like Schnirelmann density and natural density are defined. Even so it does not determine a best way to assign values to divergent sums.
  10. Jun 27, 2013 #9
    absolutely, all those things are true, which would, if this thinking were to progress, would also have such series as 1+1+1-1+1+1+1-1.... equaling zero, too. but in the Grandi's series the natural density tends towards 1/2 of either element. plus should a working solution of the Grandi's series necessarily apply to all divergent sums? or is it a possibility that this could be akin to the 180 degree triangle. where in euclidean geometry it is 180 but under conditions of positive or negative curvature the answer could be less or more. is it possible that all three answers 1, 0.5, and 0 are correct depending on the context where you place the Grandi's series.
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