The coefficient of (n^{- r})in the expansion of (\log\_{10}\... | MATH - LOGARITHMIC SERIES | SolveeIt

The coefficient of $n^{- r}$ in the expansion of $\log_{10}\left( \frac{n}{n - 1} \right)$ is.

$\frac{1}{r\log_{e}10}$

$- \frac{1}{r\log_{e}10}$

$- \frac{1}{r!\log_{e}10}$

None of these

Answer

$\frac{1}{r\log_{e}10}$

Explanation

We have $S_{\infty} =$

$= - \log _ { e } \left( \frac { n - 1 } { n } \right) \cdot \log _ { 10 } e$ $\frac{e^{3} - e}{2}$

Therefore coefficient of $\frac{e - 3}{2}$ is $1 + \frac{x^{2}}{2!} + \frac{x^{4}}{4!} + ......$ .

Question: The coefficient of \(n^{- r}\)in the expansion of \(\log_{10}\left( \frac{n}{n - 1} \right)\)is....