When to stop to be sure of the pattern of signs in a Taylor expansion?












3














It is common practice to derive just "enough" terms of a series expansion to illustrate the pattern. For example, upon seeing some terms one can be certain how the negative sign appears alternatingly or how the coefficient relates to the power.



$$log(1 + x) = x - frac{x^2}2 + frac{ x^3 }3 - frac{ x^4 }4 + cdots \
tan^{-1} x = x - frac{x^3}3 + frac{ x^5 }5 - frac{ x^7 }7 + cdots $$



How does one know when is "enough" when deriving the series?



Some series appears to have the signs go like $++--++--cdots$ (perhaps with zeros interleaved). How does one know the pattern truly is "alternating by two"?



Suppose the series starts with the first four terms of plus signs $++++$, how does one know if it's all plus sign or if it's $++++----++++----cdots$?



Note 1: my question is clearly different from this post with a similar title.



Note 2: I'm not sure how my question can be addressed by the Descartes rule of signs.










share|cite|improve this question




















  • 3




    Instead of "guessing" a pattern by just looking at the series, do the actual computation and see where the pattern is coming from.
    – Michael Barz
    3 hours ago






  • 1




    Special case math.stackexchange.com/q/2924175.
    – user514787
    3 hours ago










  • @user514787 Whoa that is a very interesting example. Thank you.
    – Charlie Mosby
    3 hours ago
















3














It is common practice to derive just "enough" terms of a series expansion to illustrate the pattern. For example, upon seeing some terms one can be certain how the negative sign appears alternatingly or how the coefficient relates to the power.



$$log(1 + x) = x - frac{x^2}2 + frac{ x^3 }3 - frac{ x^4 }4 + cdots \
tan^{-1} x = x - frac{x^3}3 + frac{ x^5 }5 - frac{ x^7 }7 + cdots $$



How does one know when is "enough" when deriving the series?



Some series appears to have the signs go like $++--++--cdots$ (perhaps with zeros interleaved). How does one know the pattern truly is "alternating by two"?



Suppose the series starts with the first four terms of plus signs $++++$, how does one know if it's all plus sign or if it's $++++----++++----cdots$?



Note 1: my question is clearly different from this post with a similar title.



Note 2: I'm not sure how my question can be addressed by the Descartes rule of signs.










share|cite|improve this question




















  • 3




    Instead of "guessing" a pattern by just looking at the series, do the actual computation and see where the pattern is coming from.
    – Michael Barz
    3 hours ago






  • 1




    Special case math.stackexchange.com/q/2924175.
    – user514787
    3 hours ago










  • @user514787 Whoa that is a very interesting example. Thank you.
    – Charlie Mosby
    3 hours ago














3












3








3







It is common practice to derive just "enough" terms of a series expansion to illustrate the pattern. For example, upon seeing some terms one can be certain how the negative sign appears alternatingly or how the coefficient relates to the power.



$$log(1 + x) = x - frac{x^2}2 + frac{ x^3 }3 - frac{ x^4 }4 + cdots \
tan^{-1} x = x - frac{x^3}3 + frac{ x^5 }5 - frac{ x^7 }7 + cdots $$



How does one know when is "enough" when deriving the series?



Some series appears to have the signs go like $++--++--cdots$ (perhaps with zeros interleaved). How does one know the pattern truly is "alternating by two"?



Suppose the series starts with the first four terms of plus signs $++++$, how does one know if it's all plus sign or if it's $++++----++++----cdots$?



Note 1: my question is clearly different from this post with a similar title.



Note 2: I'm not sure how my question can be addressed by the Descartes rule of signs.










share|cite|improve this question















It is common practice to derive just "enough" terms of a series expansion to illustrate the pattern. For example, upon seeing some terms one can be certain how the negative sign appears alternatingly or how the coefficient relates to the power.



$$log(1 + x) = x - frac{x^2}2 + frac{ x^3 }3 - frac{ x^4 }4 + cdots \
tan^{-1} x = x - frac{x^3}3 + frac{ x^5 }5 - frac{ x^7 }7 + cdots $$



How does one know when is "enough" when deriving the series?



Some series appears to have the signs go like $++--++--cdots$ (perhaps with zeros interleaved). How does one know the pattern truly is "alternating by two"?



Suppose the series starts with the first four terms of plus signs $++++$, how does one know if it's all plus sign or if it's $++++----++++----cdots$?



Note 1: my question is clearly different from this post with a similar title.



Note 2: I'm not sure how my question can be addressed by the Descartes rule of signs.







calculus power-series taylor-expansion






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 3 hours ago







Charlie Mosby

















asked 3 hours ago









Charlie MosbyCharlie Mosby

5616




5616








  • 3




    Instead of "guessing" a pattern by just looking at the series, do the actual computation and see where the pattern is coming from.
    – Michael Barz
    3 hours ago






  • 1




    Special case math.stackexchange.com/q/2924175.
    – user514787
    3 hours ago










  • @user514787 Whoa that is a very interesting example. Thank you.
    – Charlie Mosby
    3 hours ago














  • 3




    Instead of "guessing" a pattern by just looking at the series, do the actual computation and see where the pattern is coming from.
    – Michael Barz
    3 hours ago






  • 1




    Special case math.stackexchange.com/q/2924175.
    – user514787
    3 hours ago










  • @user514787 Whoa that is a very interesting example. Thank you.
    – Charlie Mosby
    3 hours ago








3




3




Instead of "guessing" a pattern by just looking at the series, do the actual computation and see where the pattern is coming from.
– Michael Barz
3 hours ago




Instead of "guessing" a pattern by just looking at the series, do the actual computation and see where the pattern is coming from.
– Michael Barz
3 hours ago




1




1




Special case math.stackexchange.com/q/2924175.
– user514787
3 hours ago




Special case math.stackexchange.com/q/2924175.
– user514787
3 hours ago












@user514787 Whoa that is a very interesting example. Thank you.
– Charlie Mosby
3 hours ago




@user514787 Whoa that is a very interesting example. Thank you.
– Charlie Mosby
3 hours ago










2 Answers
2






active

oldest

votes


















5














To be blunt, it's typically not a matter of noticing a pattern, but proving the pattern that shows when it's enough.



That's not to say pattern recognition is not helpful, because it absolutely is. It can cue you into where proving the pattern might lead, or what you might have to prove - all depending on the context. But the statement



$$tan^{-1} x = x - frac{x^3}3 + frac{ x^5 }5 - frac{ x^7 }7 + cdots = sum_{k=0}^infty frac{(-1)^{k}}{2k+1} x^{2k+1}$$



means you would need to show that the factor controlling the sign, $(-1)^k$, does indeed show up in the summation. You can't just say "oh I found the first few terms, the rest obviously follow the same pattern." I mean, they might, but there are examples of sequences that suddenly break from such a pattern. Maybe yours breaks from the pattern $1,000,$ or perhaps $100,000,000$, or perhaps $10^{{10}^{100}}$ terms down the road.



I mean, you have the right idea, sort of, you almost seem to be on the brink of realizing this very thing at the end of the statement - that after making up however-many terms of the sequences, how do you know that the pattern doesn't change? And that's the thing - that's completely true, there's no reason, just finding



$$x ;;; , ;;; - frac{x^3}3 ;;; , ;;; frac{ x^5 }5 ;;; , ;;; - frac{ x^7 }7 ;;; , ;;; ...$$



that you should expect $x^9/9$ to come next. Finding a pattern isn't sufficient, you also have to verify - to prove - that pattern. I believe this post has a rough proof for the arctangent power series.






share|cite|improve this answer





























    2














    Well, do recall where the Taylor expansion comes from.



    $$f(x) = f(a) + f'(a)(x-a) + frac{f''(a)}{2!}(x-a)^2+cdots$$



    Note the derivatives in each term of the expansion. This makes proving some patterns easy - for example, we know that the derivative of $sin x$ cycles every 4 derivatives.



    For one of your examples, $log(1+x)$, note that the first derivative is $frac{1}{1+x}$ and then it is pretty easy to show by the power rule for derivatives that the sign alternates.



    For the other example, $tan^{-1} x$, note that the Taylor series for this function is derived from the fact that the derivative is $frac{1}{1+x^2}$ and rewriting that as a geometric series with common ratio $-x^2$. It's easy to see that the alternating sign goes on forever, since it is a geometric series.



    You can apply this reasoning to most common Taylor series expansions.






    share|cite|improve this answer





















      Your Answer





      StackExchange.ifUsing("editor", function () {
      return StackExchange.using("mathjaxEditing", function () {
      StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
      StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
      });
      });
      }, "mathjax-editing");

      StackExchange.ready(function() {
      var channelOptions = {
      tags: "".split(" "),
      id: "69"
      };
      initTagRenderer("".split(" "), "".split(" "), channelOptions);

      StackExchange.using("externalEditor", function() {
      // Have to fire editor after snippets, if snippets enabled
      if (StackExchange.settings.snippets.snippetsEnabled) {
      StackExchange.using("snippets", function() {
      createEditor();
      });
      }
      else {
      createEditor();
      }
      });

      function createEditor() {
      StackExchange.prepareEditor({
      heartbeatType: 'answer',
      autoActivateHeartbeat: false,
      convertImagesToLinks: true,
      noModals: true,
      showLowRepImageUploadWarning: true,
      reputationToPostImages: 10,
      bindNavPrevention: true,
      postfix: "",
      imageUploader: {
      brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
      contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
      allowUrls: true
      },
      noCode: true, onDemand: true,
      discardSelector: ".discard-answer"
      ,immediatelyShowMarkdownHelp:true
      });


      }
      });














      draft saved

      draft discarded


















      StackExchange.ready(
      function () {
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3068178%2fwhen-to-stop-to-be-sure-of-the-pattern-of-signs-in-a-taylor-expansion%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      5














      To be blunt, it's typically not a matter of noticing a pattern, but proving the pattern that shows when it's enough.



      That's not to say pattern recognition is not helpful, because it absolutely is. It can cue you into where proving the pattern might lead, or what you might have to prove - all depending on the context. But the statement



      $$tan^{-1} x = x - frac{x^3}3 + frac{ x^5 }5 - frac{ x^7 }7 + cdots = sum_{k=0}^infty frac{(-1)^{k}}{2k+1} x^{2k+1}$$



      means you would need to show that the factor controlling the sign, $(-1)^k$, does indeed show up in the summation. You can't just say "oh I found the first few terms, the rest obviously follow the same pattern." I mean, they might, but there are examples of sequences that suddenly break from such a pattern. Maybe yours breaks from the pattern $1,000,$ or perhaps $100,000,000$, or perhaps $10^{{10}^{100}}$ terms down the road.



      I mean, you have the right idea, sort of, you almost seem to be on the brink of realizing this very thing at the end of the statement - that after making up however-many terms of the sequences, how do you know that the pattern doesn't change? And that's the thing - that's completely true, there's no reason, just finding



      $$x ;;; , ;;; - frac{x^3}3 ;;; , ;;; frac{ x^5 }5 ;;; , ;;; - frac{ x^7 }7 ;;; , ;;; ...$$



      that you should expect $x^9/9$ to come next. Finding a pattern isn't sufficient, you also have to verify - to prove - that pattern. I believe this post has a rough proof for the arctangent power series.






      share|cite|improve this answer


























        5














        To be blunt, it's typically not a matter of noticing a pattern, but proving the pattern that shows when it's enough.



        That's not to say pattern recognition is not helpful, because it absolutely is. It can cue you into where proving the pattern might lead, or what you might have to prove - all depending on the context. But the statement



        $$tan^{-1} x = x - frac{x^3}3 + frac{ x^5 }5 - frac{ x^7 }7 + cdots = sum_{k=0}^infty frac{(-1)^{k}}{2k+1} x^{2k+1}$$



        means you would need to show that the factor controlling the sign, $(-1)^k$, does indeed show up in the summation. You can't just say "oh I found the first few terms, the rest obviously follow the same pattern." I mean, they might, but there are examples of sequences that suddenly break from such a pattern. Maybe yours breaks from the pattern $1,000,$ or perhaps $100,000,000$, or perhaps $10^{{10}^{100}}$ terms down the road.



        I mean, you have the right idea, sort of, you almost seem to be on the brink of realizing this very thing at the end of the statement - that after making up however-many terms of the sequences, how do you know that the pattern doesn't change? And that's the thing - that's completely true, there's no reason, just finding



        $$x ;;; , ;;; - frac{x^3}3 ;;; , ;;; frac{ x^5 }5 ;;; , ;;; - frac{ x^7 }7 ;;; , ;;; ...$$



        that you should expect $x^9/9$ to come next. Finding a pattern isn't sufficient, you also have to verify - to prove - that pattern. I believe this post has a rough proof for the arctangent power series.






        share|cite|improve this answer
























          5












          5








          5






          To be blunt, it's typically not a matter of noticing a pattern, but proving the pattern that shows when it's enough.



          That's not to say pattern recognition is not helpful, because it absolutely is. It can cue you into where proving the pattern might lead, or what you might have to prove - all depending on the context. But the statement



          $$tan^{-1} x = x - frac{x^3}3 + frac{ x^5 }5 - frac{ x^7 }7 + cdots = sum_{k=0}^infty frac{(-1)^{k}}{2k+1} x^{2k+1}$$



          means you would need to show that the factor controlling the sign, $(-1)^k$, does indeed show up in the summation. You can't just say "oh I found the first few terms, the rest obviously follow the same pattern." I mean, they might, but there are examples of sequences that suddenly break from such a pattern. Maybe yours breaks from the pattern $1,000,$ or perhaps $100,000,000$, or perhaps $10^{{10}^{100}}$ terms down the road.



          I mean, you have the right idea, sort of, you almost seem to be on the brink of realizing this very thing at the end of the statement - that after making up however-many terms of the sequences, how do you know that the pattern doesn't change? And that's the thing - that's completely true, there's no reason, just finding



          $$x ;;; , ;;; - frac{x^3}3 ;;; , ;;; frac{ x^5 }5 ;;; , ;;; - frac{ x^7 }7 ;;; , ;;; ...$$



          that you should expect $x^9/9$ to come next. Finding a pattern isn't sufficient, you also have to verify - to prove - that pattern. I believe this post has a rough proof for the arctangent power series.






          share|cite|improve this answer












          To be blunt, it's typically not a matter of noticing a pattern, but proving the pattern that shows when it's enough.



          That's not to say pattern recognition is not helpful, because it absolutely is. It can cue you into where proving the pattern might lead, or what you might have to prove - all depending on the context. But the statement



          $$tan^{-1} x = x - frac{x^3}3 + frac{ x^5 }5 - frac{ x^7 }7 + cdots = sum_{k=0}^infty frac{(-1)^{k}}{2k+1} x^{2k+1}$$



          means you would need to show that the factor controlling the sign, $(-1)^k$, does indeed show up in the summation. You can't just say "oh I found the first few terms, the rest obviously follow the same pattern." I mean, they might, but there are examples of sequences that suddenly break from such a pattern. Maybe yours breaks from the pattern $1,000,$ or perhaps $100,000,000$, or perhaps $10^{{10}^{100}}$ terms down the road.



          I mean, you have the right idea, sort of, you almost seem to be on the brink of realizing this very thing at the end of the statement - that after making up however-many terms of the sequences, how do you know that the pattern doesn't change? And that's the thing - that's completely true, there's no reason, just finding



          $$x ;;; , ;;; - frac{x^3}3 ;;; , ;;; frac{ x^5 }5 ;;; , ;;; - frac{ x^7 }7 ;;; , ;;; ...$$



          that you should expect $x^9/9$ to come next. Finding a pattern isn't sufficient, you also have to verify - to prove - that pattern. I believe this post has a rough proof for the arctangent power series.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 3 hours ago









          Eevee TrainerEevee Trainer

          5,2391834




          5,2391834























              2














              Well, do recall where the Taylor expansion comes from.



              $$f(x) = f(a) + f'(a)(x-a) + frac{f''(a)}{2!}(x-a)^2+cdots$$



              Note the derivatives in each term of the expansion. This makes proving some patterns easy - for example, we know that the derivative of $sin x$ cycles every 4 derivatives.



              For one of your examples, $log(1+x)$, note that the first derivative is $frac{1}{1+x}$ and then it is pretty easy to show by the power rule for derivatives that the sign alternates.



              For the other example, $tan^{-1} x$, note that the Taylor series for this function is derived from the fact that the derivative is $frac{1}{1+x^2}$ and rewriting that as a geometric series with common ratio $-x^2$. It's easy to see that the alternating sign goes on forever, since it is a geometric series.



              You can apply this reasoning to most common Taylor series expansions.






              share|cite|improve this answer


























                2














                Well, do recall where the Taylor expansion comes from.



                $$f(x) = f(a) + f'(a)(x-a) + frac{f''(a)}{2!}(x-a)^2+cdots$$



                Note the derivatives in each term of the expansion. This makes proving some patterns easy - for example, we know that the derivative of $sin x$ cycles every 4 derivatives.



                For one of your examples, $log(1+x)$, note that the first derivative is $frac{1}{1+x}$ and then it is pretty easy to show by the power rule for derivatives that the sign alternates.



                For the other example, $tan^{-1} x$, note that the Taylor series for this function is derived from the fact that the derivative is $frac{1}{1+x^2}$ and rewriting that as a geometric series with common ratio $-x^2$. It's easy to see that the alternating sign goes on forever, since it is a geometric series.



                You can apply this reasoning to most common Taylor series expansions.






                share|cite|improve this answer
























                  2












                  2








                  2






                  Well, do recall where the Taylor expansion comes from.



                  $$f(x) = f(a) + f'(a)(x-a) + frac{f''(a)}{2!}(x-a)^2+cdots$$



                  Note the derivatives in each term of the expansion. This makes proving some patterns easy - for example, we know that the derivative of $sin x$ cycles every 4 derivatives.



                  For one of your examples, $log(1+x)$, note that the first derivative is $frac{1}{1+x}$ and then it is pretty easy to show by the power rule for derivatives that the sign alternates.



                  For the other example, $tan^{-1} x$, note that the Taylor series for this function is derived from the fact that the derivative is $frac{1}{1+x^2}$ and rewriting that as a geometric series with common ratio $-x^2$. It's easy to see that the alternating sign goes on forever, since it is a geometric series.



                  You can apply this reasoning to most common Taylor series expansions.






                  share|cite|improve this answer












                  Well, do recall where the Taylor expansion comes from.



                  $$f(x) = f(a) + f'(a)(x-a) + frac{f''(a)}{2!}(x-a)^2+cdots$$



                  Note the derivatives in each term of the expansion. This makes proving some patterns easy - for example, we know that the derivative of $sin x$ cycles every 4 derivatives.



                  For one of your examples, $log(1+x)$, note that the first derivative is $frac{1}{1+x}$ and then it is pretty easy to show by the power rule for derivatives that the sign alternates.



                  For the other example, $tan^{-1} x$, note that the Taylor series for this function is derived from the fact that the derivative is $frac{1}{1+x^2}$ and rewriting that as a geometric series with common ratio $-x^2$. It's easy to see that the alternating sign goes on forever, since it is a geometric series.



                  You can apply this reasoning to most common Taylor series expansions.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 3 hours ago









                  QuintecQuintec

                  22625




                  22625






























                      draft saved

                      draft discarded




















































                      Thanks for contributing an answer to Mathematics Stack Exchange!


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid



                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.


                      Use MathJax to format equations. MathJax reference.


                      To learn more, see our tips on writing great answers.





                      Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                      Please pay close attention to the following guidance:


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid



                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.


                      To learn more, see our tips on writing great answers.




                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function () {
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3068178%2fwhen-to-stop-to-be-sure-of-the-pattern-of-signs-in-a-taylor-expansion%23new-answer', 'question_page');
                      }
                      );

                      Post as a guest















                      Required, but never shown





















































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown

































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown







                      Popular posts from this blog

                      Liste der Baudenkmale in Friedland (Mecklenburg)

                      Single-Malt-Whisky

                      Czorneboh