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Find $lim_nto infty left(left(1+frac 1n2^n+1right)^n -1right)*2^n$.

0

$begingroup$

I know the limit is $frac 12$ and I proved it with sandwich theorem. Does anyone know how to prove it in a constructive way.

$$lim_nto infty left(left(1+frac 1n2^n+1right)^n -1right)*2^n$$

sequences-and-series limits

share|cite|improve this question

edited Apr 8 at 19:35

Shaun

10.5k113687

asked Apr 8 at 19:27

Alessandro CignaAlessandro Cigna

113

$endgroup$

add a comment | 

0

$begingroup$

I know the limit is $frac 12$ and I proved it with sandwich theorem. Does anyone know how to prove it in a constructive way.

$$lim_nto infty left(left(1+frac 1n2^n+1right)^n -1right)*2^n$$

sequences-and-series limits

share|cite|improve this question

edited Apr 8 at 19:35

Shaun

10.5k113687

asked Apr 8 at 19:27

Alessandro CignaAlessandro Cigna

113

$endgroup$

add a comment | 

$begingroup$

I know the limit is $frac 12$ and I proved it with sandwich theorem. Does anyone know how to prove it in a constructive way.

$$lim_nto infty left(left(1+frac 1n2^n+1right)^n -1right)*2^n$$

sequences-and-series limits

share|cite|improve this question

edited Apr 8 at 19:35

Shaun

10.5k113687

asked Apr 8 at 19:27

Alessandro CignaAlessandro Cigna

113

$endgroup$

share|cite|improve this question

edited Apr 8 at 19:35

Shaun

10.5k113687

asked Apr 8 at 19:27

Alessandro CignaAlessandro Cigna

113

share|cite|improve this question

edited Apr 8 at 19:35

Shaun

10.5k113687

asked Apr 8 at 19:27

Alessandro CignaAlessandro Cigna

113

edited Apr 8 at 19:35

Shaun

10.5k113687

edited Apr 8 at 19:35

Shaun

10.5k113687

asked Apr 8 at 19:27

Alessandro CignaAlessandro Cigna

113

asked Apr 8 at 19:27

Alessandro CignaAlessandro Cigna

113

1 Answer
1

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1

$begingroup$

Hint $:$ Expand binomially

$$left (1 + frac 1 n2^n right )^n.$$

You will see that all the terms of $left(left(1+frac 1n2^n+1right)^n -1right)*2^n$ will vanish when $n rightarrow infty$ except that $frac 1 2.$

share|cite|improve this answer

edited Apr 8 at 19:42

answered Apr 8 at 19:33

Dbchatto67Dbchatto67

3,052623

$endgroup$

add a comment | 

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1

$begingroup$

Hint $:$ Expand binomially

$$left (1 + frac 1 n2^n right )^n.$$

You will see that all the terms of $left(left(1+frac 1n2^n+1right)^n -1right)*2^n$ will vanish when $n rightarrow infty$ except that $frac 1 2.$

share|cite|improve this answer

edited Apr 8 at 19:42

answered Apr 8 at 19:33

Dbchatto67Dbchatto67

3,052623

$endgroup$

add a comment | 

1

$begingroup$

Hint $:$ Expand binomially

$$left (1 + frac 1 n2^n right )^n.$$

You will see that all the terms of $left(left(1+frac 1n2^n+1right)^n -1right)*2^n$ will vanish when $n rightarrow infty$ except that $frac 1 2.$

share|cite|improve this answer

edited Apr 8 at 19:42

answered Apr 8 at 19:33

Dbchatto67Dbchatto67

3,052623

$endgroup$

add a comment | 

$begingroup$

Hint $:$ Expand binomially

$$left (1 + frac 1 n2^n right )^n.$$

You will see that all the terms of $left(left(1+frac 1n2^n+1right)^n -1right)*2^n$ will vanish when $n rightarrow infty$ except that $frac 1 2.$

share|cite|improve this answer

edited Apr 8 at 19:42

answered Apr 8 at 19:33

Dbchatto67Dbchatto67

3,052623

$endgroup$

share|cite|improve this answer

edited Apr 8 at 19:42

answered Apr 8 at 19:33

Dbchatto67Dbchatto67

3,052623

answered Apr 8 at 19:33

Dbchatto67Dbchatto67

3,052623

answered Apr 8 at 19:33

Dbchatto67Dbchatto67

3,052623