03Zcalculatinge

= Calculating e =

{Not part of the course}

Euler's Number, e, is an irrational number.

math e \approx 2.7182818 \dots math

e can be calculated in a number of ways. Here are two of them.

Method 1
math e = \begin{matrix} \lim \\ k \rightarrow \infty \\ \end{matrix} \left( 1 + \dfrac{1}{k} \right)^k math

ie

math k = 2 \qquad \qquad e \approx \left( 1 + \dfrac{1}{2} \right)^2 = \left( \dfrac{3}{2} \right)^2 = 2.25 math . math k = 3 \qquad \qquad e \approx \left( 1 + \dfrac{1}{3} \right)^3 = \left( \dfrac{4}{3} \right)^3 = 2.37037037 math . math k = 4 \qquad \qquad e \approx \left( 1 + \dfrac{1}{4} \right)^4 = \left( \dfrac{5}{4} \right)^4 = 2.44140625 math

etc

This is best done in a spreadsheet: In a similar way, we can find the value of e x for any value of x
 * Note: **

math e^x = \begin{matrix} \lim \\ k \rightarrow \infty \\ \end{matrix} \left( 1 + \dfrac{x}{k} \right)^k math

Method 2
math e = 1 + \begin{matrix} \lim \\ k \rightarrow \infty \\ \end{matrix} \sum\limits_{n=1}^{k} { \dfrac{1}{n!} } = 1 + \dfrac{1}{1!} + \dfrac{1}{2!} + \dfrac{1}{3!} + \dfrac{1}{4!} + \dots math

ie

math k = 2 \qquad \qquad e \approx 1 + \dfrac{1}{1!} + \dfrac{1}{2!} = 1 + 1 + \dfrac{1}{2} = 2.5 math . math k = 3 \qquad \qquad e \approx 1 + \dfrac{1}{1!} + \dfrac{1}{2!} + \dfrac{1}{3!} = 1 + 1 + \dfrac{1}{2} + \dfrac{1}{6} = 2.6666667 math . math k = 4 \qquad \qquad e \approx 1 + \dfrac{1}{1!} + \dfrac{1}{2!} + \dfrac{1}{3!} + \dfrac{1}{4!} = 1 + 1 + \dfrac{1}{2} + \dfrac{1}{6} + \dfrac{1}{24} = 2.7083333 math

etc

Again, using a spreadsheet:

This formula is clearly much more efficient. It approaches the exact value of e much more quickly. After 12 iterations it has calculated e correct to 9 decimal places. The previous method took 10,000,000 iterations to get e accurate to 6 decimal places.

In a similar way, we can find the value of e x for any value of x
 * Note **

math e^x = 1 + \begin{matrix} \lim \\ k \rightarrow \infty \\ \end{matrix} \sum\limits_{n=1}^{k} { \dfrac{x}{n!} } = 1 + \dfrac{x}{1!} + \dfrac{x}{2!} + \dfrac{x}{3!} + \dfrac{x}{4!} + \dots math

Of course, the quickest way to calculate e is to put e^1 on your calculator!!!
 * Note **

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