Chapter 4.3, Problem 51E

Solution

To calculate: The value of $n$ at which the given expression produces the value of $e$ at nine decimal place.

  $n={10}^{10}$

Given:

An expression ${\left(1+\frac{1}{n}\right)}^n$ is given.

Concept used:

Substitute the different values of $n$ in the expression and check at which value of $n$ the expression ${\left(1+\frac{1}{n}\right)}^n$ produces the correct value of $e$ at $9$ decimal places.

Calculation:

Consider the following table which is generated using the table feature of the graphing calculator and shows the calculation:

$n$	${\left(1+\frac{1}{n}\right)}^n$
${10}^1$	$2.59374$
${10}^2$	$2.70481$
${10}^3$	$2.71692$
${10}^4$	$2.71815$
${10}^5$	$2.71827$
${10}^6$	$2.71828$
${10}^7$	$2.718281$
${10}^8$	$2.7182818$
${10}^9$	$2.71828182$
${10}^{10}$	$2.718281828$

So, for $n={10}^{10}$ , ${\left(1+\frac{1}{n}\right)}^n$ gives the correct value of $e$ at $9$ decimal places.

The small value of $n$ does not make significant changes in the values produced by the ${\left(1+\frac{1}{n}\right)}^n$ but the larger values take one step closer to the result and it can be observed in the table.

Conclusion:

The value of $n$ is ${10}^{10}$ and the procedure of obtaining the result is to check the expression output at different value of $n$ .