Sum of a series of a number raised to incrementing powers












4












$begingroup$


How would I estimate the sum of a series of numbers like this: $2^0+2^1+2^2+2^3+cdots+2^n$. What math course deals with this sort of calculation? Thanks much!






sequences-and-series







share|cite|improve this question











$endgroup$








  • 4




    $begingroup$
    Two great courses would be real analysis and Calculus.
    $endgroup$
    – Julian Rachman
    Dec 28 '14 at 0:29








  • 1




    $begingroup$
    it is called Geometric progession. see here: en.wikipedia.org/wiki/Geometric_progression
    $endgroup$
    – Krish
    Dec 28 '14 at 0:31






  • 2




    $begingroup$
    Actually, for finite sums like this, the course would be algebra (in the sense of "high-school algebra", or "college algebra", not "abstract algebra" or "modern algebra").
    $endgroup$
    – GEdgar
    Dec 28 '14 at 0:42






  • 1




    $begingroup$
    @GEdgar You are correct. But it is only "spoken of" in high school, not understood like in Real Analysis. It is essentially the foundation of Calculus in itself containing a section specifically for "sequences and series."
    $endgroup$
    – Julian Rachman
    Dec 28 '14 at 0:47










  • $begingroup$
    $$ sum_{n=0}^{infty text{ or finite number}} 2^n $$ I think Calculus or some class like advanced algebra or college algebra
    $endgroup$
    – Irrational Person
    Dec 28 '14 at 0:53


















4












$begingroup$


How would I estimate the sum of a series of numbers like this: $2^0+2^1+2^2+2^3+cdots+2^n$. What math course deals with this sort of calculation? Thanks much!






sequences-and-series







share|cite|improve this question











$endgroup$








  • 4




    $begingroup$
    Two great courses would be real analysis and Calculus.
    $endgroup$
    – Julian Rachman
    Dec 28 '14 at 0:29








  • 1




    $begingroup$
    it is called Geometric progession. see here: en.wikipedia.org/wiki/Geometric_progression
    $endgroup$
    – Krish
    Dec 28 '14 at 0:31






  • 2




    $begingroup$
    Actually, for finite sums like this, the course would be algebra (in the sense of "high-school algebra", or "college algebra", not "abstract algebra" or "modern algebra").
    $endgroup$
    – GEdgar
    Dec 28 '14 at 0:42






  • 1




    $begingroup$
    @GEdgar You are correct. But it is only "spoken of" in high school, not understood like in Real Analysis. It is essentially the foundation of Calculus in itself containing a section specifically for "sequences and series."
    $endgroup$
    – Julian Rachman
    Dec 28 '14 at 0:47










  • $begingroup$
    $$ sum_{n=0}^{infty text{ or finite number}} 2^n $$ I think Calculus or some class like advanced algebra or college algebra
    $endgroup$
    – Irrational Person
    Dec 28 '14 at 0:53
















4












4








4


2



$begingroup$


How would I estimate the sum of a series of numbers like this: $2^0+2^1+2^2+2^3+cdots+2^n$. What math course deals with this sort of calculation? Thanks much!






sequences-and-series







share|cite|improve this question











$endgroup$




How would I estimate the sum of a series of numbers like this: $2^0+2^1+2^2+2^3+cdots+2^n$. What math course deals with this sort of calculation? Thanks much!






sequences-and-series




sequences-and-series






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 28 '14 at 4:43









Michael Hardy

1




1










asked Dec 28 '14 at 0:27









Julian A.Julian A.

170115




170115








  • 4




    $begingroup$
    Two great courses would be real analysis and Calculus.
    $endgroup$
    – Julian Rachman
    Dec 28 '14 at 0:29








  • 1




    $begingroup$
    it is called Geometric progession. see here: en.wikipedia.org/wiki/Geometric_progression
    $endgroup$
    – Krish
    Dec 28 '14 at 0:31






  • 2




    $begingroup$
    Actually, for finite sums like this, the course would be algebra (in the sense of "high-school algebra", or "college algebra", not "abstract algebra" or "modern algebra").
    $endgroup$
    – GEdgar
    Dec 28 '14 at 0:42






  • 1




    $begingroup$
    @GEdgar You are correct. But it is only "spoken of" in high school, not understood like in Real Analysis. It is essentially the foundation of Calculus in itself containing a section specifically for "sequences and series."
    $endgroup$
    – Julian Rachman
    Dec 28 '14 at 0:47










  • $begingroup$
    $$ sum_{n=0}^{infty text{ or finite number}} 2^n $$ I think Calculus or some class like advanced algebra or college algebra
    $endgroup$
    – Irrational Person
    Dec 28 '14 at 0:53
















  • 4




    $begingroup$
    Two great courses would be real analysis and Calculus.
    $endgroup$
    – Julian Rachman
    Dec 28 '14 at 0:29








  • 1




    $begingroup$
    it is called Geometric progession. see here: en.wikipedia.org/wiki/Geometric_progression
    $endgroup$
    – Krish
    Dec 28 '14 at 0:31






  • 2




    $begingroup$
    Actually, for finite sums like this, the course would be algebra (in the sense of "high-school algebra", or "college algebra", not "abstract algebra" or "modern algebra").
    $endgroup$
    – GEdgar
    Dec 28 '14 at 0:42






  • 1




    $begingroup$
    @GEdgar You are correct. But it is only "spoken of" in high school, not understood like in Real Analysis. It is essentially the foundation of Calculus in itself containing a section specifically for "sequences and series."
    $endgroup$
    – Julian Rachman
    Dec 28 '14 at 0:47










  • $begingroup$
    $$ sum_{n=0}^{infty text{ or finite number}} 2^n $$ I think Calculus or some class like advanced algebra or college algebra
    $endgroup$
    – Irrational Person
    Dec 28 '14 at 0:53










4




4




$begingroup$
Two great courses would be real analysis and Calculus.
$endgroup$
– Julian Rachman
Dec 28 '14 at 0:29






$begingroup$
Two great courses would be real analysis and Calculus.
$endgroup$
– Julian Rachman
Dec 28 '14 at 0:29






1




1




$begingroup$
it is called Geometric progession. see here: en.wikipedia.org/wiki/Geometric_progression
$endgroup$
– Krish
Dec 28 '14 at 0:31




$begingroup$
it is called Geometric progession. see here: en.wikipedia.org/wiki/Geometric_progression
$endgroup$
– Krish
Dec 28 '14 at 0:31




2




2




$begingroup$
Actually, for finite sums like this, the course would be algebra (in the sense of "high-school algebra", or "college algebra", not "abstract algebra" or "modern algebra").
$endgroup$
– GEdgar
Dec 28 '14 at 0:42




$begingroup$
Actually, for finite sums like this, the course would be algebra (in the sense of "high-school algebra", or "college algebra", not "abstract algebra" or "modern algebra").
$endgroup$
– GEdgar
Dec 28 '14 at 0:42




1




1




$begingroup$
@GEdgar You are correct. But it is only "spoken of" in high school, not understood like in Real Analysis. It is essentially the foundation of Calculus in itself containing a section specifically for "sequences and series."
$endgroup$
– Julian Rachman
Dec 28 '14 at 0:47




$begingroup$
@GEdgar You are correct. But it is only "spoken of" in high school, not understood like in Real Analysis. It is essentially the foundation of Calculus in itself containing a section specifically for "sequences and series."
$endgroup$
– Julian Rachman
Dec 28 '14 at 0:47












$begingroup$
$$ sum_{n=0}^{infty text{ or finite number}} 2^n $$ I think Calculus or some class like advanced algebra or college algebra
$endgroup$
– Irrational Person
Dec 28 '14 at 0:53






$begingroup$
$$ sum_{n=0}^{infty text{ or finite number}} 2^n $$ I think Calculus or some class like advanced algebra or college algebra
$endgroup$
– Irrational Person
Dec 28 '14 at 0:53












3 Answers
3






active

oldest

votes


















7












$begingroup$

Two great math courses that deal with sums and sequences such as the one you have defined are Real Analysis and basic Single-Variable Calculus. This problem specifically deals with geometric progression. Yes, you do learn some in high school, but not that much. Real Analysis is a subject that gives you a more structured intuition for these types of problems.



The solution to your problem is this by a geometric sum:



$$2^0+2^1+2^2+2^3+cdotcdotcdot+2^n=frac{2^{n+1}-1}{2-1}=boxed{2^{n+1}-1}.$$






share|cite|improve this answer











$endgroup$













  • $begingroup$
    @Julian Thank you for accepting my answer. If you take a look at my blog, I have a great reference under the "Notes" page by a professor at the University of Louisville. Just jump to the section on sequences (I believe Chapter 3). You gain a better intuition for these types of questions.
    $endgroup$
    – Julian Rachman
    Dec 28 '14 at 0:48










  • $begingroup$
    :) Thanks! I'll check it out.
    $endgroup$
    – Julian A.
    Dec 28 '14 at 8:04










  • $begingroup$
    I learnt this in calculus in high school; it's a simple geometric progression. There's no need to go into real analysis for such basic things. It'll simply be too much.
    $endgroup$
    – user122283
    Jan 14 '15 at 3:23





















3












$begingroup$

There is no need to estimate, the exact answer is $2^{n+1}-1$.



This is basic algebra as taught in schools.






share|cite|improve this answer









$endgroup$





















    2












    $begingroup$

    That is a geometric sum:



    $${ 2 }^{ 0 }+{ 2 }^{ 1 }+{ 2 }^{ 2 }+{ 2 }^{ 3 }+...+{ 2 }^{ n }=frac{2^{n+1}-1}{2-1}$$



    It is taught at highschools.






    share|cite|improve this answer









    $endgroup$














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      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      7












      $begingroup$

      Two great math courses that deal with sums and sequences such as the one you have defined are Real Analysis and basic Single-Variable Calculus. This problem specifically deals with geometric progression. Yes, you do learn some in high school, but not that much. Real Analysis is a subject that gives you a more structured intuition for these types of problems.



      The solution to your problem is this by a geometric sum:



      $$2^0+2^1+2^2+2^3+cdotcdotcdot+2^n=frac{2^{n+1}-1}{2-1}=boxed{2^{n+1}-1}.$$






      share|cite|improve this answer











      $endgroup$













      • $begingroup$
        @Julian Thank you for accepting my answer. If you take a look at my blog, I have a great reference under the "Notes" page by a professor at the University of Louisville. Just jump to the section on sequences (I believe Chapter 3). You gain a better intuition for these types of questions.
        $endgroup$
        – Julian Rachman
        Dec 28 '14 at 0:48










      • $begingroup$
        :) Thanks! I'll check it out.
        $endgroup$
        – Julian A.
        Dec 28 '14 at 8:04










      • $begingroup$
        I learnt this in calculus in high school; it's a simple geometric progression. There's no need to go into real analysis for such basic things. It'll simply be too much.
        $endgroup$
        – user122283
        Jan 14 '15 at 3:23


















      7












      $begingroup$

      Two great math courses that deal with sums and sequences such as the one you have defined are Real Analysis and basic Single-Variable Calculus. This problem specifically deals with geometric progression. Yes, you do learn some in high school, but not that much. Real Analysis is a subject that gives you a more structured intuition for these types of problems.



      The solution to your problem is this by a geometric sum:



      $$2^0+2^1+2^2+2^3+cdotcdotcdot+2^n=frac{2^{n+1}-1}{2-1}=boxed{2^{n+1}-1}.$$






      share|cite|improve this answer











      $endgroup$













      • $begingroup$
        @Julian Thank you for accepting my answer. If you take a look at my blog, I have a great reference under the "Notes" page by a professor at the University of Louisville. Just jump to the section on sequences (I believe Chapter 3). You gain a better intuition for these types of questions.
        $endgroup$
        – Julian Rachman
        Dec 28 '14 at 0:48










      • $begingroup$
        :) Thanks! I'll check it out.
        $endgroup$
        – Julian A.
        Dec 28 '14 at 8:04










      • $begingroup$
        I learnt this in calculus in high school; it's a simple geometric progression. There's no need to go into real analysis for such basic things. It'll simply be too much.
        $endgroup$
        – user122283
        Jan 14 '15 at 3:23
















      7












      7








      7





      $begingroup$

      Two great math courses that deal with sums and sequences such as the one you have defined are Real Analysis and basic Single-Variable Calculus. This problem specifically deals with geometric progression. Yes, you do learn some in high school, but not that much. Real Analysis is a subject that gives you a more structured intuition for these types of problems.



      The solution to your problem is this by a geometric sum:



      $$2^0+2^1+2^2+2^3+cdotcdotcdot+2^n=frac{2^{n+1}-1}{2-1}=boxed{2^{n+1}-1}.$$






      share|cite|improve this answer











      $endgroup$



      Two great math courses that deal with sums and sequences such as the one you have defined are Real Analysis and basic Single-Variable Calculus. This problem specifically deals with geometric progression. Yes, you do learn some in high school, but not that much. Real Analysis is a subject that gives you a more structured intuition for these types of problems.



      The solution to your problem is this by a geometric sum:



      $$2^0+2^1+2^2+2^3+cdotcdotcdot+2^n=frac{2^{n+1}-1}{2-1}=boxed{2^{n+1}-1}.$$







      share|cite|improve this answer














      share|cite|improve this answer



      share|cite|improve this answer








      edited Dec 22 '18 at 8:01









      Carl Schildkraut

      11.9k11444




      11.9k11444










      answered Dec 28 '14 at 0:35









      Julian RachmanJulian Rachman

      1,227923




      1,227923












      • $begingroup$
        @Julian Thank you for accepting my answer. If you take a look at my blog, I have a great reference under the "Notes" page by a professor at the University of Louisville. Just jump to the section on sequences (I believe Chapter 3). You gain a better intuition for these types of questions.
        $endgroup$
        – Julian Rachman
        Dec 28 '14 at 0:48










      • $begingroup$
        :) Thanks! I'll check it out.
        $endgroup$
        – Julian A.
        Dec 28 '14 at 8:04










      • $begingroup$
        I learnt this in calculus in high school; it's a simple geometric progression. There's no need to go into real analysis for such basic things. It'll simply be too much.
        $endgroup$
        – user122283
        Jan 14 '15 at 3:23




















      • $begingroup$
        @Julian Thank you for accepting my answer. If you take a look at my blog, I have a great reference under the "Notes" page by a professor at the University of Louisville. Just jump to the section on sequences (I believe Chapter 3). You gain a better intuition for these types of questions.
        $endgroup$
        – Julian Rachman
        Dec 28 '14 at 0:48










      • $begingroup$
        :) Thanks! I'll check it out.
        $endgroup$
        – Julian A.
        Dec 28 '14 at 8:04










      • $begingroup$
        I learnt this in calculus in high school; it's a simple geometric progression. There's no need to go into real analysis for such basic things. It'll simply be too much.
        $endgroup$
        – user122283
        Jan 14 '15 at 3:23


















      $begingroup$
      @Julian Thank you for accepting my answer. If you take a look at my blog, I have a great reference under the "Notes" page by a professor at the University of Louisville. Just jump to the section on sequences (I believe Chapter 3). You gain a better intuition for these types of questions.
      $endgroup$
      – Julian Rachman
      Dec 28 '14 at 0:48




      $begingroup$
      @Julian Thank you for accepting my answer. If you take a look at my blog, I have a great reference under the "Notes" page by a professor at the University of Louisville. Just jump to the section on sequences (I believe Chapter 3). You gain a better intuition for these types of questions.
      $endgroup$
      – Julian Rachman
      Dec 28 '14 at 0:48












      $begingroup$
      :) Thanks! I'll check it out.
      $endgroup$
      – Julian A.
      Dec 28 '14 at 8:04




      $begingroup$
      :) Thanks! I'll check it out.
      $endgroup$
      – Julian A.
      Dec 28 '14 at 8:04












      $begingroup$
      I learnt this in calculus in high school; it's a simple geometric progression. There's no need to go into real analysis for such basic things. It'll simply be too much.
      $endgroup$
      – user122283
      Jan 14 '15 at 3:23






      $begingroup$
      I learnt this in calculus in high school; it's a simple geometric progression. There's no need to go into real analysis for such basic things. It'll simply be too much.
      $endgroup$
      – user122283
      Jan 14 '15 at 3:23













      3












      $begingroup$

      There is no need to estimate, the exact answer is $2^{n+1}-1$.



      This is basic algebra as taught in schools.






      share|cite|improve this answer









      $endgroup$


















        3












        $begingroup$

        There is no need to estimate, the exact answer is $2^{n+1}-1$.



        This is basic algebra as taught in schools.






        share|cite|improve this answer









        $endgroup$
















          3












          3








          3





          $begingroup$

          There is no need to estimate, the exact answer is $2^{n+1}-1$.



          This is basic algebra as taught in schools.






          share|cite|improve this answer









          $endgroup$



          There is no need to estimate, the exact answer is $2^{n+1}-1$.



          This is basic algebra as taught in schools.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 28 '14 at 0:32









          Suzu HiroseSuzu Hirose

          4,18021228




          4,18021228























              2












              $begingroup$

              That is a geometric sum:



              $${ 2 }^{ 0 }+{ 2 }^{ 1 }+{ 2 }^{ 2 }+{ 2 }^{ 3 }+...+{ 2 }^{ n }=frac{2^{n+1}-1}{2-1}$$



              It is taught at highschools.






              share|cite|improve this answer









              $endgroup$


















                2












                $begingroup$

                That is a geometric sum:



                $${ 2 }^{ 0 }+{ 2 }^{ 1 }+{ 2 }^{ 2 }+{ 2 }^{ 3 }+...+{ 2 }^{ n }=frac{2^{n+1}-1}{2-1}$$



                It is taught at highschools.






                share|cite|improve this answer









                $endgroup$
















                  2












                  2








                  2





                  $begingroup$

                  That is a geometric sum:



                  $${ 2 }^{ 0 }+{ 2 }^{ 1 }+{ 2 }^{ 2 }+{ 2 }^{ 3 }+...+{ 2 }^{ n }=frac{2^{n+1}-1}{2-1}$$



                  It is taught at highschools.






                  share|cite|improve this answer









                  $endgroup$



                  That is a geometric sum:



                  $${ 2 }^{ 0 }+{ 2 }^{ 1 }+{ 2 }^{ 2 }+{ 2 }^{ 3 }+...+{ 2 }^{ n }=frac{2^{n+1}-1}{2-1}$$



                  It is taught at highschools.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 28 '14 at 0:31









                  ajotatxeajotatxe

                  54.1k24190




                  54.1k24190






























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