NZ Level 8 (NZC) Level 3 (NCEA) [In development]

Natural Logarithms

The natural base $e$`e` (Euler’s number) is defined as $e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$`e`=lim`n`→∞(1+1`n`)`n`

The table shows the value of $\left(1+\frac{1}{n}\right)^n$(1+1`n`)`n` using various values of $n$`n`.

$n$n |
$\left(1+\frac{1}{n}\right)^n$(1+1n)n |
---|---|

$1$1 | $2$2 |

$100$100 | $1.01^{100}=2.704813$1.01100=2.704813 ... |

$1000$1000 | $1.001^{1000}=2.716923$1.0011000=2.716923 ... |

$10000$10000 | $1.0001^{10000}=2.718145$1.000110000=2.718145 ... |

$100000$100000 | $1.00001^{100000}=2.718268$1.00001100000=2.718268 ... |

a

Evaluate $\left(1+\frac{1}{n}\right)^n$(1+1`n`)`n` for $n=1000000$`n`=1000000, correct to six decimal places.

b

Which of the following is the closest approximation of $e$`e`?

$2.718280821$2.718280821

A

$2.718281828$2.718281828

B

$2.718281820$2.718281820

C

$2.718281818$2.718281818

D

$2.718280821$2.718280821

A

$2.718281828$2.718281828

B

$2.718281820$2.718281820

C

$2.718281818$2.718281818

D

Easy

Approx 3 minutes

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