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BEAUTIFUL - Beautiful patterns in Math

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  • #16
    Originally posted by inertia View Post

    Since Euler did not place specific boundary conditions on it, I'll hold off from here.

    I'm convinced of it's divergence based on L'Hopital's rule as shown above. Maybe Euler was trying to get his students to think instead of appealing to his authority.

    Maybe the following rearrangement will make this preposterous statement easier to visualize:

    ...+ 1/x3 + 1/x2 + 1/x = - ( 1 + x + x2 + x3 +...)

    The limit x --> infinity on the rhs is -(infinity) by visual inspection alone. On the lhs the limit x --> infinity is zero by visual inspection too.

    So, saying zero is equal to infinity is well...(fill in the blank )
    I would sure like to look at it in your book. But unfortunately the link that you gave me only directs me to where I might purchase the book.

    Comment


    • #17
      Originally posted by cisco qid View Post

      I would sure like to look at it in your book. But unfortunately the link that you gave me only directs me to where I might purchase the book.
      Here you go: https://avionicsengineering.files.wo...-finney-9e.pdf

      Comment


      • #18
        Originally posted by cisco qid View Post

        I would sure like to look at it in your book. But unfortunately the link that you gave me only directs me to where I might purchase the book.
        Here is the quote from a later version: https://books.google.com/books?id=mX...ts+limitations
        "The exact sciences also start from the assumption that in the end it will always be possible to understand nature, even in every new field of experience, but that we make no a priori assumptions about the meaning of the word "understand"."

        Heisenberg
        .....................

        "The heavens declare the glory of God; the skies proclaim the work of his hands. Day after day they pour forth speech; night after night they reveal knowledge." ( Psalm 19:1-2 )

        Comment


        • #19
          Originally posted by Pragmatic View Post

          When I examine graphs that contain a series like (1/xn + 1/xn+1...) if the n is odd then the graph indicates that y = -1 when x = -1. If n is even then y = 0 when x = -1. While I am unsure how to prove this, the trend I see indicates that if (x + x2...) = 0 then you would be left with (-1) + 1 + (0) = 0. I am no mathematician so I am ready to be corrected on this.
          Interesting

          For the series in question ...+ 1/x3 + 1/x2 + 1/x +1 + x + x2 + x3 +... = 0 , it states that any x-value you choose will always sum to zero. If it is correct, then any x you choose should always provide a y =0, always.

          A "p series test" helps for part of the series*.

          For the form: ...+ 1/xp + 1/xp + 1/x +1 ..., it will always converge if p > 1 and will diverge if p is less than or equal to 1. The p-term for 1/x is 1.

          Cisco qid's correct assessment of a power series like: 1 + x + x2 + x3 +..., is useful where rules for domain provide boundary conditions.

          Example: 1/(1-x) = 1 + x + x2 + x3 +..., with a domain of (-1 < x < 1) works.

          However, for the series in question neither of these approaches is adequate to analyze its behavior. Your result using n is odd or even is interesting, but I believe that it does not provide enough information. Maybe I will try a graphical approach too.

          * See: https://math.oregonstate.edu/home/pr.../p-series.html
          Last edited by inertia; 01-30-18, 03:13 PM.
          "The exact sciences also start from the assumption that in the end it will always be possible to understand nature, even in every new field of experience, but that we make no a priori assumptions about the meaning of the word "understand"."

          Heisenberg
          .....................

          "The heavens declare the glory of God; the skies proclaim the work of his hands. Day after day they pour forth speech; night after night they reveal knowledge." ( Psalm 19:1-2 )

          Comment


          • #20
            Originally posted by inertia View Post

            Interesting

            For the series in question ...+ 1/x3 + 1/x2 + 1/x +1 + x + x2 + x3 +... = 0 , it states that any x-value you choose will always sum to zero. If it is correct, then any x you choose should always provide a y =0, always.

            A "p series test" helps for part of the series*.

            For the form: ...+ 1/xp + 1/xp + 1/x +1 ..., it will always converge if p > 1 and will diverge if p is less than or equal to 1. The p-term for 1/x is 1.

            Cisco qid's correct assessment of a power series like: 1 + x + x2 + x3 +..., is useful where rules for domain provide boundary conditions.

            Example: 1/(1-x) = 1 + x + x2 + x3 +..., with a domain of (-1 < x < 1) works.

            However, for the series in question neither of these approaches is adequate to analyze its behavior. Your result using n is odd or even is interesting, but I believe that it does not provide enough information. Maybe I will try a graphical approach too.

            * See: https://math.oregonstate.edu/home/pr.../p-series.html
            Not to beat a dead horse to death but if you want to see something really weird - you can actually derive the expression,


            ...+ 1/x3 + 1/x2 + 1/x +1 + x + x2 + x3 +... = 0, as follows:


            Start with the expression:

            S = 1/x + 1/x2 + 1/x3

            Then

            S = 1/x(1 + 1/x + 1/x2 + ) = 1/x(1 + S) = 1/x + S/x

            or

            S - S/x = 1/x => S = 1/(x 1) = -1/(1-x) = - (1 + x + x2 + x3 ....) = 1/x + 1/x2 + 1/x3

            And we get out result which we know is false since,

            1/x + 1/x2 + 1/x3 = 1/(x 1) can't be true because x = 0 is not allowed on the LHS while it is allowed on the RHS.

            Comment


            • #21
              Originally posted by cisco qid View Post

              Not to beat a dead horse to death but if you want to see something really weird - you can actually derive the expression,


              ...+ 1/x3 + 1/x2 + 1/x +1 + x + x2 + x3 +... = 0, as follows:


              Start with the expression:

              S = 1/x + 1/x2 + 1/x3

              Then

              S = 1/x(1 + 1/x + 1/x2 + …) = 1/x(1 + S) = 1/x + S/x

              or

              S - S/x = 1/x => S = 1/(x – 1) = -1/(1-x) = - (1 + x + x2 + x3 ....) = 1/x + 1/x2 + 1/x3

              And we get out result which we know is false since,

              1/x + 1/x2 + 1/x3 … = 1/(x – 1) can't be true because x = 0 is not allowed on the LHS while it is allowed on the RHS.
              It is undefined but if you graph the left (make the right y) then the trend is that if n is odd in (1/x​​​​​​n + 1/x n+1+...) then x approaches -1 as y ​​​approaches -1.

              If the trend is true then we simply show that (x + x2 + x3 ....) is 0 when x is -1 and we are left via substitution with (0 + 1 - 1 = 0).

              I am no mathematician so if you find an error above then keep that in mind.
              If God did not exist, it would be necessary to invent him. - Voltaire

              Comment


              • #22
                Originally posted by cisco qid View Post

                Not to beat a dead horse to death but if you want to see something really weird - you can actually derive the expression,


                ...+ 1/x3 + 1/x2 + 1/x +1 + x + x2 + x3 +... = 0, as follows:


                Start with the expression:

                S = 1/x + 1/x2 + 1/x3

                Then

                S = 1/x(1 + 1/x + 1/x2 + …) = 1/x(1 + S) = 1/x + S/x

                or

                S - S/x = 1/x => S = 1/(x – 1) = -1/(1-x) = - (1 + x + x2 + x3 ....) = 1/x + 1/x2 + 1/x3

                And we get out result which we know is false since,

                1/x + 1/x2 + 1/x3 … = 1/(x – 1) can't be true because x = 0 is not allowed on the LHS while it is allowed on the RHS.
                Nice!
                "The exact sciences also start from the assumption that in the end it will always be possible to understand nature, even in every new field of experience, but that we make no a priori assumptions about the meaning of the word "understand"."

                Heisenberg
                .....................

                "The heavens declare the glory of God; the skies proclaim the work of his hands. Day after day they pour forth speech; night after night they reveal knowledge." ( Psalm 19:1-2 )

                Comment


                • #23
                  Originally posted by Pragmatic View Post

                  It is undefined but if you graph the left (make the right y) then the trend is that if n is odd in (1/x​​​​​​n + 1/x n+1+...) then x approaches -1 as y ​​​approaches -1.

                  If the trend is true then we simply show that (x + x2 + x3 ....) is 0 when x is -1 and we are left via substitution with (0 + 1 - 1 = 0).

                  I am no mathematician so if you find an error above then keep that in mind.
                  I know I can't find an error because I don't know how to graph and infinite series.

                  Comment


                  • #24
                    Originally posted by cisco qid View Post

                    Not to beat a dead horse to death but if you want to see something really weird - you can actually derive the expression,


                    ...+ 1/x3 + 1/x2 + 1/x +1 + x + x2 + x3 +... = 0, as follows:


                    Start with the expression:

                    S = 1/x + 1/x2 + 1/x3

                    Then

                    S = 1/x(1 + 1/x + 1/x2 + …) = 1/x(1 + S) = 1/x + S/x

                    or

                    S - S/x = 1/x => S = 1/(x – 1) = -1/(1-x) = - (1 + x + x2 + x3 ....) = 1/x + 1/x2 + 1/x3

                    And we get out result which we know is false since,

                    1/x + 1/x2 + 1/x3 … = 1/(x – 1) can't be true because x = 0 is not allowed on the LHS while it is allowed on the RHS.
                    That's in effect what I wrote several posts ago. But I won't charge you franchise fees



                    Regards, HRG.

                    "The universe doesn't care what happens to its inhabitants, but its inhabitants do" (Tyrrho).

                    Comment


                    • #25
                      Originally posted by HRG View Post

                      That's in effect what I wrote several posts ago. But I won't charge you franchise fees


                      What you stated was Sx = S and since x is arbitrary S must be zero, which we all now know is false, while I derived the original series. Charging for someone else's intellectual property is copyright infringement.
                      Last edited by cisco qid; 02-12-18, 06:31 PM.

                      Comment


                      • #26
                        Originally posted by cisco qid View Post

                        What you stated was Sx = S and since x is arbitrary S must be zero, which we all now know is false, while I derived the original series. Charging for someone else's intellectual property is copyright infringement.
                        If x is arbitrary (i.e. a variable), then we can insert 4 for X. From 4S = S we can infer that S = 0. Thus it's you who infringed my intellectual property.
                        Regards, HRG.

                        "The universe doesn't care what happens to its inhabitants, but its inhabitants do" (Tyrrho).

                        Comment


                        • #27
                          Originally posted by HRG View Post

                          If x is arbitrary (i.e. a variable), then we can insert 4 for X. From 4S = S we can infer that S = 0. Thus it's you who infringed my intellectual property.
                          I believe Euler would have objected since he stated it first.

                          Comment


                          • #28
                            Originally posted by Pragmatic View Post

                            When I examine graphs that contain a series like (1/xn + 1/xn+1...) if the n is odd then the graph indicates that y = -1 when x = -1. If n is even then y = 0 when x = -1. While I am unsure how to prove this, the trend I see indicates that if (x + x2...) = 0 then you would be left with (-1) + 1 + (0) = 0. I am no mathematician so I am ready to be corrected on this.
                            Here is what I get when modeled by symbolic Matlab up to the tenth power:


                            sum(1 + x + x^2 + x^3 + x^4 + x^5+ x^6 + x^7 + x^8 + x^9 + x^10+ 1/x + 1/x^2 + 1/x^3 + 1/x^4 + 1/x^5 + 1/x^6 + 1/x^7 + 1/x^8 + 1/x^9 + 1/x^10,x=1..infinity )

                            infinity

                            Trying again starting at zero:


                            sum(1 + x + x^2 + x^3 + x^4 + x^5+ x^6 + x^7 + x^8 + x^9 + x^10+ 1/x + 1/x^2 + 1/x^3 + 1/x^4 + 1/x^5 + 1/x^6 + 1/x^7 + 1/x^8 + 1/x^9 + 1/x^10,x=0..infinity )

                            infinity

                            Trying again from - infinity to + infinity


                            sum(1 + x + x^2 + x^3 + x^4 + x^5+ x^6 + x^7 + x^8 + x^9 + x^10+ 1/x + 1/x^2 + 1/x^3 + 1/x^4 + 1/x^5 + 1/x^6 + 1/x^7 + 1/x^8 + 1/x^9 + 1/x^10,x=-infinity..infinity )

                            infinity




                            "The exact sciences also start from the assumption that in the end it will always be possible to understand nature, even in every new field of experience, but that we make no a priori assumptions about the meaning of the word "understand"."

                            Heisenberg
                            .....................

                            "The heavens declare the glory of God; the skies proclaim the work of his hands. Day after day they pour forth speech; night after night they reveal knowledge." ( Psalm 19:1-2 )

                            Comment


                            • #29
                              Originally posted by inertia View Post

                              Here is what I get when modeled by symbolic Matlab up to the tenth power:


                              sum(1 + x + x^2 + x^3 + x^4 + x^5+ x^6 + x^7 + x^8 + x^9 + x^10+ 1/x + 1/x^2 + 1/x^3 + 1/x^4 + 1/x^5 + 1/x^6 + 1/x^7 + 1/x^8 + 1/x^9 + 1/x^10,x=1..infinity )

                              infinity
                              I also found this result interesting: From x = -500 to 500


                              sum(1 + x + x^2 + x^3 + x^4 + x^5+ x^6 + x^7 + x^8 + x^9 + x^10+ 1/x + 1/x^2 + 1/x^3 + 1/x^4 + 1/x^5 + 1/x^6 + 1/x^7 + 1/x^8 + 1/x^9 + 1/x^10,x = -500..500)
                              Error: Singularity. [sum::sum]

                              This next one makes sense: From x = -1 to 1


                              sum(1 + x + x^2 + x^3 + x^4 + x^5+ x^6 + x^7 + x^8 + x^9 + x^10+ 1/x + 1/x^2 + 1/x^3 + 1/x^4 + 1/x^5 + 1/x^6 + 1/x^7 + 1/x^8 + 1/x^9 + 1/x^10,x = -1..1)
                              Error: Division by zero. [_power]
                              Evaluating: sum::sum

                              Getting a result that = 0 was...err...difficult indeed. As expected....
                              "The exact sciences also start from the assumption that in the end it will always be possible to understand nature, even in every new field of experience, but that we make no a priori assumptions about the meaning of the word "understand"."

                              Heisenberg
                              .....................

                              "The heavens declare the glory of God; the skies proclaim the work of his hands. Day after day they pour forth speech; night after night they reveal knowledge." ( Psalm 19:1-2 )

                              Comment


                              • #30
                                Originally posted by Pragmatic View Post

                                It is undefined but if you graph the left (make the right y) then the trend is that if n is odd in (1/x​​​​​​n + 1/x n+1+...) then x approaches -1 as y ​​​approaches -1.

                                If the trend is true then we simply show that (x + x2 + x3 ....) is 0 when x is -1 and we are left via substitution with (0 + 1 - 1 = 0).

                                I am no mathematician so if you find an error above then keep that in mind.
                                Here is what I get when I graph the equation:



                                It is easy to see why the equation can never provide a sum equaling zero. Thanks for the graphing idea.
                                "The exact sciences also start from the assumption that in the end it will always be possible to understand nature, even in every new field of experience, but that we make no a priori assumptions about the meaning of the word "understand"."

                                Heisenberg
                                .....................

                                "The heavens declare the glory of God; the skies proclaim the work of his hands. Day after day they pour forth speech; night after night they reveal knowledge." ( Psalm 19:1-2 )

                                Comment

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