The value of the series $x \log _e a+\frac{x^3}{3 !}\left(\log _e a\right)^3$ $+\frac{x^5}{5 !}\left(\log _e…
The value of the series $x \log _e a+\frac{x^3}{3 !}\left(\log _e a\right)^3$ $+\frac{x^5}{5 !}\left(\log _e a\right)^5+\ldots$ is
- $\cos h\left(x \log _e a\right)$
- $\cot h\left(x \log _e a\right)$
- $\sin h\left(x \log _e a\right)$
- $\tan h\left(x \log _e a\right)$
Solution
We have a series
$
\begin{aligned}
& x \log _e a+\frac{x^3}{3 !}\left(\log _e a\right)^3+\frac{x^5}{5 !}\left(\log _e a\right)^5+\ldots \\
&=\frac{e^{x \log _e a}-e^{-x \log _e a}}{2} \\
& {\left[\because \frac{e^x-e^{-x}}{2}=x+\frac{x^3}{3 !}+\frac{x^5}{5 !}+\ldots\right] } \\
&= \sin h\left(x \log _e a\right)
\end{aligned}
$
Asked in: AP EAMCET 2004
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