\[\frac{x - 1}{(x + 1)} + \frac{1}{2}.\frac{x^{2} - 1}{(x +... | MATH - LOGARITHMIC SERIES | SolveeIt

Question

$\frac{x - 1}{(x + 1)} + \frac{1}{2}.\frac{x^{2} - 1}{(x + 1)^{2}} + \frac{1}{3}.\frac{x^{3} - 1}{(x + 1)^{3}} + ......\infty =$

$\log_{e}x$

$\log_{e}(1 + x)$

$\log_{e}(1 - x)$

$\log_{e}\frac{x}{1 + x}$

Answer

$\log_{e}x$

Explanation

Solution

$\frac{( - 1)^{n + 1}}{n}\lbrack 2^{n} + 1\rbrack$

$\frac{2^{n} + 1}{n}$

$= - \log _ { e } \left( 1 - \frac { x } { x + 1 } \right) - \left\{ - \log _ { e } \left( 1 - \frac { 1 } { x + 1 } \right) \right\}$ $\sum_{x = 1}^{1999}{\log_{n}x}$ .

Trick : Put $\sqrt[1999]{1999}$ and check

Question: \[\frac{x - 1}{(x + 1)} + \frac{1}{2}.\frac{x^{2} - 1}{(x + 1)^{2}} + \frac{1}{3}.\frac{x^{3} - 1}{(...

Solution