Limit of sequence using Euler's sequence

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$lim_ntoinfty left(1+frac12n+3right)^n$



I know that this approaches $e^1/2$ however don't know how to prove this. Any hints are appreciated.




sequences-and-series limits




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  • The way you have types the expression the limit is $1$. I believe you wanted the n-th power outside $()$.
    – Kavi Rama Murthy
    Aug 14 at 9:03











  • Do you know that $(1+(1/n))^nto e$? If so then this implies that $(1+(1/(2n+3)))^2n+3to e$.
    – Paramanand Singh
    Aug 14 at 9:08














up vote
0
down vote

favorite












$lim_ntoinfty left(1+frac12n+3right)^n$



I know that this approaches $e^1/2$ however don't know how to prove this. Any hints are appreciated.




sequences-and-series limits




share|cite|improve this question






















  • The way you have types the expression the limit is $1$. I believe you wanted the n-th power outside $()$.
    – Kavi Rama Murthy
    Aug 14 at 9:03











  • Do you know that $(1+(1/n))^nto e$? If so then this implies that $(1+(1/(2n+3)))^2n+3to e$.
    – Paramanand Singh
    Aug 14 at 9:08












up vote
0
down vote

favorite









up vote
0
down vote

favorite











$lim_ntoinfty left(1+frac12n+3right)^n$



I know that this approaches $e^1/2$ however don't know how to prove this. Any hints are appreciated.




sequences-and-series limits




share|cite|improve this question














$lim_ntoinfty left(1+frac12n+3right)^n$



I know that this approaches $e^1/2$ however don't know how to prove this. Any hints are appreciated.





sequences-and-series limits





share|cite|improve this question













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share|cite|improve this question








edited Aug 14 at 9:10









Paramanand Singh

45.3k553142




45.3k553142










asked Aug 14 at 9:00









RJM

133




133











  • The way you have types the expression the limit is $1$. I believe you wanted the n-th power outside $()$.
    – Kavi Rama Murthy
    Aug 14 at 9:03











  • Do you know that $(1+(1/n))^nto e$? If so then this implies that $(1+(1/(2n+3)))^2n+3to e$.
    – Paramanand Singh
    Aug 14 at 9:08
















  • The way you have types the expression the limit is $1$. I believe you wanted the n-th power outside $()$.
    – Kavi Rama Murthy
    Aug 14 at 9:03











  • Do you know that $(1+(1/n))^nto e$? If so then this implies that $(1+(1/(2n+3)))^2n+3to e$.
    – Paramanand Singh
    Aug 14 at 9:08















The way you have types the expression the limit is $1$. I believe you wanted the n-th power outside $()$.
– Kavi Rama Murthy
Aug 14 at 9:03





The way you have types the expression the limit is $1$. I believe you wanted the n-th power outside $()$.
– Kavi Rama Murthy
Aug 14 at 9:03













Do you know that $(1+(1/n))^nto e$? If so then this implies that $(1+(1/(2n+3)))^2n+3to e$.
– Paramanand Singh
Aug 14 at 9:08




Do you know that $(1+(1/n))^nto e$? If so then this implies that $(1+(1/(2n+3)))^2n+3to e$.
– Paramanand Singh
Aug 14 at 9:08










3 Answers
3






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3
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Hint: $left(1+frac12n+3right)^2n+3=left(left(1+frac12n+3right)^nright)^2left(1+frac12n+3right)^3$.






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  • Simple and clear. +1
    – Paramanand Singh
    Aug 14 at 9:11










  • @ Paramanand: thanks !
    – Fred
    Aug 14 at 9:13










  • Very ingenious! +1
    – Davide Morgante
    Aug 14 at 9:15

















up vote
1
down vote













Simply put
$$
2n+3=miff n=fracm -32
$$
Now we have
$$
lim_ntoinftyleft(1+frac12n+3right)^n= lim_mtoinftyleft(1+frac1mright)^fracm -32
$$
and applying the usual rules of calculation of limits you get the sought for result.






share|cite|improve this answer



























    up vote
    0
    down vote













    $$lim_ntoinftyleft(1+dfrac1an+bright)^n=left(lim_ntoinftyleft(1+dfrac1an+bright)^an+bright)^lim_ntoinftydfrac nan+b$$






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      3 Answers
      3






      active

      oldest

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      3 Answers
      3






      active

      oldest

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      active

      oldest

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      active

      oldest

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      up vote
      3
      down vote













      Hint: $left(1+frac12n+3right)^2n+3=left(left(1+frac12n+3right)^nright)^2left(1+frac12n+3right)^3$.






      share|cite|improve this answer




















      • Simple and clear. +1
        – Paramanand Singh
        Aug 14 at 9:11










      • @ Paramanand: thanks !
        – Fred
        Aug 14 at 9:13










      • Very ingenious! +1
        – Davide Morgante
        Aug 14 at 9:15














      up vote
      3
      down vote













      Hint: $left(1+frac12n+3right)^2n+3=left(left(1+frac12n+3right)^nright)^2left(1+frac12n+3right)^3$.






      share|cite|improve this answer




















      • Simple and clear. +1
        – Paramanand Singh
        Aug 14 at 9:11










      • @ Paramanand: thanks !
        – Fred
        Aug 14 at 9:13










      • Very ingenious! +1
        – Davide Morgante
        Aug 14 at 9:15












      up vote
      3
      down vote










      up vote
      3
      down vote









      Hint: $left(1+frac12n+3right)^2n+3=left(left(1+frac12n+3right)^nright)^2left(1+frac12n+3right)^3$.






      share|cite|improve this answer












      Hint: $left(1+frac12n+3right)^2n+3=left(left(1+frac12n+3right)^nright)^2left(1+frac12n+3right)^3$.







      share|cite|improve this answer












      share|cite|improve this answer



      share|cite|improve this answer










      answered Aug 14 at 9:09









      Fred

      37.9k1238




      37.9k1238











      • Simple and clear. +1
        – Paramanand Singh
        Aug 14 at 9:11










      • @ Paramanand: thanks !
        – Fred
        Aug 14 at 9:13










      • Very ingenious! +1
        – Davide Morgante
        Aug 14 at 9:15
















      • Simple and clear. +1
        – Paramanand Singh
        Aug 14 at 9:11










      • @ Paramanand: thanks !
        – Fred
        Aug 14 at 9:13










      • Very ingenious! +1
        – Davide Morgante
        Aug 14 at 9:15















      Simple and clear. +1
      – Paramanand Singh
      Aug 14 at 9:11




      Simple and clear. +1
      – Paramanand Singh
      Aug 14 at 9:11












      @ Paramanand: thanks !
      – Fred
      Aug 14 at 9:13




      @ Paramanand: thanks !
      – Fred
      Aug 14 at 9:13












      Very ingenious! +1
      – Davide Morgante
      Aug 14 at 9:15




      Very ingenious! +1
      – Davide Morgante
      Aug 14 at 9:15










      up vote
      1
      down vote













      Simply put
      $$
      2n+3=miff n=fracm -32
      $$
      Now we have
      $$
      lim_ntoinftyleft(1+frac12n+3right)^n= lim_mtoinftyleft(1+frac1mright)^fracm -32
      $$
      and applying the usual rules of calculation of limits you get the sought for result.






      share|cite|improve this answer
























        up vote
        1
        down vote













        Simply put
        $$
        2n+3=miff n=fracm -32
        $$
        Now we have
        $$
        lim_ntoinftyleft(1+frac12n+3right)^n= lim_mtoinftyleft(1+frac1mright)^fracm -32
        $$
        and applying the usual rules of calculation of limits you get the sought for result.






        share|cite|improve this answer






















          up vote
          1
          down vote










          up vote
          1
          down vote









          Simply put
          $$
          2n+3=miff n=fracm -32
          $$
          Now we have
          $$
          lim_ntoinftyleft(1+frac12n+3right)^n= lim_mtoinftyleft(1+frac1mright)^fracm -32
          $$
          and applying the usual rules of calculation of limits you get the sought for result.






          share|cite|improve this answer












          Simply put
          $$
          2n+3=miff n=fracm -32
          $$
          Now we have
          $$
          lim_ntoinftyleft(1+frac12n+3right)^n= lim_mtoinftyleft(1+frac1mright)^fracm -32
          $$
          and applying the usual rules of calculation of limits you get the sought for result.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Aug 14 at 9:19









          Daniele Tampieri

          7571313




          7571313




















              up vote
              0
              down vote













              $$lim_ntoinftyleft(1+dfrac1an+bright)^n=left(lim_ntoinftyleft(1+dfrac1an+bright)^an+bright)^lim_ntoinftydfrac nan+b$$






              share|cite|improve this answer
























                up vote
                0
                down vote













                $$lim_ntoinftyleft(1+dfrac1an+bright)^n=left(lim_ntoinftyleft(1+dfrac1an+bright)^an+bright)^lim_ntoinftydfrac nan+b$$






                share|cite|improve this answer






















                  up vote
                  0
                  down vote










                  up vote
                  0
                  down vote









                  $$lim_ntoinftyleft(1+dfrac1an+bright)^n=left(lim_ntoinftyleft(1+dfrac1an+bright)^an+bright)^lim_ntoinftydfrac nan+b$$






                  share|cite|improve this answer












                  $$lim_ntoinftyleft(1+dfrac1an+bright)^n=left(lim_ntoinftyleft(1+dfrac1an+bright)^an+bright)^lim_ntoinftydfrac nan+b$$







                  share|cite|improve this answer












                  share|cite|improve this answer



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                  answered Aug 14 at 9:08









                  lab bhattacharjee

                  215k14152264




                  215k14152264






















                       

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