Announcement

Collapse

Natural Science 301 Guidelines

This is an open forum area for all members for discussions on all issues of science and origins. This area will and does get volatile at times, but we ask that it be kept to a dull roar, and moderators will intervene to keep the peace if necessary. This means obvious trolling and flaming that becomes a problem will be dealt with, and you might find yourself in the doghouse.

As usual, Tweb rules apply. If you haven't read them now would be a good time.

Forum Rules: Here
See more
See less

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

Collapse
X
 
  • Filter
  • Time
  • Show
Clear All
new posts

  • 1 + 2 + 3 +... = -1/12

    In another thread the prospect of infinite sums came up and I wanted to introduce one of the oddest results I've ever seen in math. Its a situation where clearly the partial sums tend towards infinite, and where its possible to develops means of summing up the infinite numbers in such a way that it returns a specific result.

    Let's dive right into it and give the result.



    This ridiculously counterintuitive result stems from a field called Ramanujan summanation, which is one in a group of summation methods for infinite series that are divergent. Strangely enough some of these methods can be used to sum up the infinite series of integers, and they always end up given the result above.

    I'll show you how this result can be derived from something called Abel summation.

    Now if we take another series, namely



    Then naturally we'd try to solve this by finding its partial sum (summing up all but the leading term) and seeing what this converges to



    Obviously the partial sum is divergent as we add more and more terms. So we'll need a different way of summing. The mathematician Abel proposed (though Euler found it first), the following means of handling this type of summation.



    This can't sum the first series mentioned, but it is able to sum the second one, giving us...



    From this we can determine what series mentioned in the beginning will be, if it has any value at all. This is done simple be substracting the partial sum of one series from the other.



    Of course this result can also be found directly, but that requires a stronger method of summation than Abel summation, such as Ramanujan summation. However unsurprisingly it yields the exact same answer. And its not merely a theoretical answer, as sums over all the natural integers occur often in quantum field theory, and in the derivation of the Casimir effect one has to use such a sum.

    Therefore this result, along with various strong derivations are included in Advanced Quantum Mechanics course work.

    What do you guys think?

  • #2
    I'm of the opinion that it is a special case. Modifying the sum, how it is derived. That being said:

    . . . the Gospel of Christ, for it is [the] power of God to salvation to every [one] believing, . . . -- Romans 1:16.

    . . . that Christ died for our sins according to the scriptures; And that he was buried, and that he rose again the third day according to the scriptures: . . . -- 1 Corinthians 15:3, 4.

    Whosoever believeth that Jesus is the Christ is born of God: . . . -- 1 John 5:1.

    Comment


    • #3
      Yes you can't do it with all series, there are series such that all the summation methods that we know will fail to make a definite result of them. And really it shouldn't be thought of as a proper summation. Results such as -1/12 should be thought of as representing a property of the partial sums, rather than really representing the sum of all integers.

      Comment


      • #4
        Originally posted by Leonhard View Post
        Yes you can't do it with all series, there are series such that all the summation methods that we know will fail to make a definite result of them. And really it shouldn't be thought of as a proper summation. Results such as -1/12 should be thought of as representing a property of the partial sums, rather than really representing the sum of all integers.
        Yet it is still the limit result of an infinite series taken it that special why. And it seems to be a relevant part in String Theory.
        . . . the Gospel of Christ, for it is [the] power of God to salvation to every [one] believing, . . . -- Romans 1:16.

        . . . that Christ died for our sins according to the scriptures; And that he was buried, and that he rose again the third day according to the scriptures: . . . -- 1 Corinthians 15:3, 4.

        Whosoever believeth that Jesus is the Christ is born of God: . . . -- 1 John 5:1.

        Comment


        • #5
          Originally posted by 37818 View Post
          Yet it is still the limit result of an infinite series taken it that special why. And it seems to be a relevant part in String Theory.
          So far there's no realistic interpretation of why applying what's called 'regulators' works out in quantum field theory (not just in String Theory). Its a mystery that has yet to be completely solved, but there's work in progress on it.

          Personally I'm not fond of String Theory, but for other reasons I won't go into.

          Comment


          • #6
            See the exchange between lao tzu and me that starts here: http://www.theologyweb.com/campus/sh...ll=1#post45840

            I think this result is caused by the artificial introduction of randomness into the 'idea' of infinity.
            βλέπομεν γὰρ ἄρτι δι᾿ ἐσόπτρου ἐν αἰνίγματι, τότε δὲ πρόσωπον πρὸς πρόσωπον
            ἄρτι γινώσκω ἐκ μέρους, τότε δὲ ἐπιγνώσομαι καθὼς καὶ ἐπεγνώσθην.

            אָכֵ֕ן אַתָּ֖ה אֵ֣ל מִסְתַּתֵּ֑ר אֱלֹהֵ֥י יִשְׂרָאֵ֖ל מוֹשִֽׁיעַ׃

            Comment


            • #7
              Originally posted by robrecht View Post
              See the exchange between lao tzu and me that starts here: http://www.theologyweb.com/campus/sh...ll=1#post45840

              I think this result is caused by the artificial introduction of randomness into the 'idea' of infinity.
              Where and how is the randomness introduced? A fixed pattern is hardly random.
              . . . the Gospel of Christ, for it is [the] power of God to salvation to every [one] believing, . . . -- Romans 1:16.

              . . . that Christ died for our sins according to the scriptures; And that he was buried, and that he rose again the third day according to the scriptures: . . . -- 1 Corinthians 15:3, 4.

              Whosoever believeth that Jesus is the Christ is born of God: . . . -- 1 John 5:1.

              Comment


              • #8
                Does this mean that if you travel with infinite speed you start moving back in time?
                "As for my people, children are their oppressors, and women rule over them. O my people, they which lead thee cause thee to err, and destroy the way of thy paths." Isaiah 3:12

                There is no such thing as innocence, only degrees of guilt.

                Comment


                • #9
                  Originally posted by 37818 View Post
                  Where and how is the randomness introduced? A fixed pattern is hardly random.
                  Watch the video, starting around 2 minutes. The sum of S1, depending on where you stop, will either be one or zero. Since they cannot determine what number will be stopped at, they take the average, which is 1/2. That's randomness. In 'reality', if you are counting an infinite number of numbers, you do not stop anywhere, but keep going on forever.
                  βλέπομεν γὰρ ἄρτι δι᾿ ἐσόπτρου ἐν αἰνίγματι, τότε δὲ πρόσωπον πρὸς πρόσωπον
                  ἄρτι γινώσκω ἐκ μέρους, τότε δὲ ἐπιγνώσομαι καθὼς καὶ ἐπεγνώσθην.

                  אָכֵ֕ן אַתָּ֖ה אֵ֣ל מִסְתַּתֵּ֑ר אֱלֹהֵ֥י יִשְׂרָאֵ֖ל מוֹשִֽׁיעַ׃

                  Comment


                  • #10
                    Originally posted by robrecht View Post
                    Watch the video, starting around 2 minutes. The sum of S1, depending on where you stop, will either be one or zero. Since they cannot determine what number will be stopped at, they take the average, which is 1/2. That's randomness. In 'reality', if you are counting an infinite number of numbers, you do not stop anywhere, but keep going on forever.
                    That's not randomness. That's taking an average over a field of values. Nothing random about it.
                    "[Mathematics] is the revealer of every genuine truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start he will never make his entry through the portals of wisdom."
                    --Thomas Bradwardine, De Continuo (c. 1325)

                    Comment


                    • #11
                      Originally posted by robrecht View Post
                      Watch the video, starting around 2 minutes. The sum of S1, depending on where you stop, will either be one or zero. Since they cannot determine what number will be stopped at, they take the average, which is 1/2. That's randomness. In 'reality', if you are counting an infinite number of numbers, you do not stop anywhere, but keep going on forever.
                      Maybe 'ad hoc' is a better word to describe what you mean than randomness?

                      Comment


                      • #12
                        Originally posted by Boxing Pythagoras View Post
                        That's not randomness. That's taking an average over a field of values. Nothing random about it.
                        An average based on probability rather an arbitrary choice, which in 'reality' should not be made in a sum of a 'truly infinite' number of numbers.
                        βλέπομεν γὰρ ἄρτι δι᾿ ἐσόπτρου ἐν αἰνίγματι, τότε δὲ πρόσωπον πρὸς πρόσωπον
                        ἄρτι γινώσκω ἐκ μέρους, τότε δὲ ἐπιγνώσομαι καθὼς καὶ ἐπεγνώσθην.

                        אָכֵ֕ן אַתָּ֖ה אֵ֣ל מִסְתַּתֵּ֑ר אֱלֹהֵ֥י יִשְׂרָאֵ֖ל מוֹשִֽׁיעַ׃

                        Comment


                        • #13
                          Originally posted by Leonhard View Post
                          Maybe 'ad hoc' is a better word to describe what you mean than randomness?
                          Perhaps. I'm not sure of what technical definitions of 'randomness' might or might not be used here. What seems random or arbitrary is that the average is based on one or another number (one or zero) depending on where in the series one might 'stop', with a 50% chance of either sum being the result. This seems artificially introduced into the idea of infinity, according to a mathematical attempt to come to terms with infinity.
                          βλέπομεν γὰρ ἄρτι δι᾿ ἐσόπτρου ἐν αἰνίγματι, τότε δὲ πρόσωπον πρὸς πρόσωπον
                          ἄρτι γινώσκω ἐκ μέρους, τότε δὲ ἐπιγνώσομαι καθὼς καὶ ἐπεγνώσθην.

                          אָכֵ֕ן אַתָּ֖ה אֵ֣ל מִסְתַּתֵּ֑ר אֱלֹהֵ֥י יִשְׂרָאֵ֖ל מוֹשִֽׁיעַ׃

                          Comment


                          • #14
                            Originally posted by robrecht View Post
                            Perhaps. I'm not sure of what technical definitions of 'randomness' might or might not be used here. What seems random or arbitrary is that the average is based on one or another number (one or zero) depending on where in the series one might 'stop', with a 50% chance of either sum being the result. This seems artificially introduced into the idea of infinity, according to a mathematical attempt to come to terms with infinity.
                            Oh it has nothing to with infinity as such! Its just that when you have an infinite series, there's more than one way of summing it up. You're quite correct that the partial sum diverges (jumping between 1 and 0), and so you can't sum up the infinite series by looking at the limit of the partial sum. Which is what you'd typically do.

                            Instead what you could do is you introduce a regulator, which is what is done in Abel summation. Its a small factor multiplied to each member of the series, and then we look at the limit as this factor goes towards one. This way of summing up the series produces the answer 1/2.

                            Comment


                            • #15
                              Originally posted by Leonhard View Post
                              Now if we take another series, namely



                              Then naturally we'd try to solve this by finding its partial sum (summing up all but the leading term) and seeing what this converges to

                              You're not actually summing up all but the leading term the way you've shown it. The summation would go to negative infinity given that each value subtracted is higher than the number added directly before it: (1-2)+(3-4)+(4-5)... With each pair, you're shifting the summation further negative by 1.


                              Originally posted by Leonhard View Post
                              Obviously the partial sum is divergent as we add more and more terms. So we'll need a different way of summing. The mathematician Abel proposed (though Euler found it first), the following means of handling this type of summation.



                              This can't sum the first series mentioned, but it is able to sum the second one, giving us...

                              I'd hardly say the partial sum is 'obviously' divergent, but I'm not really worried about that part here. If a proposed solution for summing integers results in a fraction, I'd say that solution isn't any solution at all. Abel's summation might work where the integers are positive, but it seems to not apply when those integers are negative. You've also made up a completely new series and taken to calling it part of the original series.


                              Originally posted by Leonhard View Post
                              From this we can determine what series mentioned in the beginning will be, if it has any value at all. This is done simple be substracting the partial sum of one series from the other.



                              Of course this result can also be found directly, but that requires a stronger method of summation than Abel summation, such as Ramanujan summation. However unsurprisingly it yields the exact same answer. And its not merely a theoretical answer, as sums over all the natural integers occur often in quantum field theory, and in the derivation of the Casimir effect one has to use such a sum.

                              Therefore this result, along with various strong derivations are included in Advanced Quantum Mechanics course work.

                              What do you guys think?
                              This makes no sense to me. You created a second series, used a summation method that seems to give an obviously false result, and then proceeded to subtract the second series from the first in an attempt to solve its summation.
                              I'm not here anymore.

                              Comment

                              Related Threads

                              Collapse

                              Topics Statistics Last Post
                              Started by Juvenal, 11-30-2020, 04:47 PM
                              2 responses
                              28 views
                              1 like
                              Last Post Seeker
                              by Seeker
                               
                              Started by rogue06, 11-28-2020, 12:54 PM
                              4 responses
                              37 views
                              0 likes
                              Last Post shunyadragon  
                              Started by shunyadragon, 11-26-2020, 09:46 PM
                              0 responses
                              12 views
                              1 like
                              Last Post shunyadragon  
                              Started by lee_merrill, 11-23-2020, 10:25 PM
                              5 responses
                              50 views
                              1 like
                              Last Post Seeker
                              by Seeker
                               
                              Started by rogue06, 11-22-2020, 08:25 AM
                              5 responses
                              74 views
                              3 likes
                              Last Post rogue06
                              by rogue06
                               
                              Working...
                              X