User:Pulimaiyi/maths

The identity
The infinite product of the ratios of every odd-th and even-th roots of e (Euler's number) is equal to 2.
 * $$\frac{ {e} }{ \sqrt{e} } \cdot \frac{ \sqrt[3]{e}}{ \sqrt[4]{e} } \cdot \frac{ \sqrt[5]{e}}{ \sqrt[6]{e} } \cdot \frac{ \sqrt[7]{e}}{ \sqrt[8]{e} } \cdots = 2$$

It can be written more elegantly as:
 * $$\prod_{n=1}^{\infty} \frac{\sqrt[2n-1]{e}}{\sqrt[2n]{e}} = 2$$

Derivation
The natural logarithm (ln) of any number (say y) is the solution to the equation
 * $$e^x = y$$.

Then, $$ln (y) = x$$. The natural logarithm of 2 is 0.69314718056..., a transcendental number. It has an infinite series representation, which is given below:
 * $$\sum_{n = 1}^\infty \frac{(-1)^{n + 1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots\ = ln 2. $$

This is the alternating harmonic series which converges to ln (2). Since $$e^{ln(2)} = 2$$, it can also be said that $$e^{1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots} = 2$$ Addition of exponents denotes the multiplication of the base and subtraction denotes the division of the base. This allows us to write the expression $$e^{ln(2)}$$ as the following:
 * $$\frac{ {e^1} }{e^\frac{1}{2}} \cdot \frac{ {e^\frac{1}{3}}}{ {e^\frac{1}{4}} } \cdot \frac{ {e^\frac{1}{5}}}{ {e^\frac{1}{6}} } \cdot \frac{ {e^\frac{1}{7}}}{ {e^\frac{1}{8}} } \cdots$$

Any number raised to an exponent is equal to the root of the number with the reciprocal of the exponent as the index. This allows us to write the above infinite expression as:
 * $$\frac{ {e} }{ \sqrt{e} } \cdot \frac{ \sqrt[3]{e}}{ \sqrt[4]{e} } \cdot \frac{ \sqrt[5]{e}}{ \sqrt[6]{e} } \cdot \frac{ \sqrt[7]{e}}{ \sqrt[8]{e} } \cdots = e^{ln(2)} = 2$$

Almost equal to
I have found certain expressions and values to be incredibly close to other more significant values. For instance: $$ 0.4^{0.4} \approx ln(2) $$ The approximation of the natural logarithm of 2 as $$ 0.4^{0.4}$$ is unbelievably good. It is correct up to 5 decimal places. It differs from the true value of ln(2) by less than 0.00034% as shown below: $$ ln(2) = 0.69314718056...$$ $$ 0.4^{0.4} = 0.69314484315...$$