How to get the approximation of $\ln 2$ and prove results using the knowledge of senior high school(China)See in the textbook.(the figure accurate to the third decimal place)? Above the question,everything we are supposed to use(without proving):
- derivative and its main formulas to find a function's derivative
- definite integral
- Newton-Leibniz formula
In fact,there's an orignal question about it.See it in the picture.
So,I find a method to calculate it.But it is slowly to use actually.And the official answer is so ingenious to construct a fit inequality. I use the definite integral.
Since$$lnx=\int\frac{1}xdx$$ then according to the Newton-Leibniz formula:$$ln2=ln2-ln1=\int_1^2 \frac{1}xdx$$ Then the interval 1 to 2 is then divided into ten parts and approximated using the connection of the points of $\frac{1}x$ at the endpoints of the interval.It's hard to calculate and its approximation only accurate to the second decimal place.
You can use Newton's Method for approximating roots of an equation. Consider:
$f(x) = e^x - 2$
When $x = \ln(2)$, $f(x) = 0$, so:
Make a guess, say $x_0=1$, then Newton's Method says:
$x_{n+1}=x_n-f(x_n)/f'(x_n)$
$x_1 = x_0 - f(x_0)/f'(x_0)$ $= 1 - (e-2)/e$
Then:
$x_2 = x_1 - f(x_1)/f'(x_1)$
When you get to $x_3$ it's accurate to 6 decimal places ($x_2$ is accurate to 2).
Alternatively, you can use the power series representation of $\ln(1+x)$:
$\ln(1+x) = x-x^2/2+x^3/3-x^4/4+...$
Evaluating at $x=1$ gives you a rational approximation, but it's only accurate to three decimal places after 500 terms