3. Exponential & Logarithmic Functions
United States of America
Grades 9-12
Investigation: Defining Euler's number (e)
Interactive practice questions
The natural base $e$e (Euler’s number) is defined as $e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$e=limn→∞(1+1n)n
The table shows the value of $\left(1+\frac{1}{n}\right)^n$(1+1n)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+1n)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
Easy
2min
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