Question
The value of ${\log _3}\,4{\log _4}\,5{\log _5}\,6{\log _6}\,7{\log _7}\,8{\log _8}\,9$ is
$ = {{\log 4} \over {\log 3}}.{{\log 5} \over {\log 4}}.{{\log 6} \over {\log 5}}.{{\log 7} \over {\log 6}}.{{\log 8} \over {\log 7}}.{{\log 9} \over {\log 8}} = {{\log 9} \over {\log 3}}$
$ = {\log _3}9 = {\log _3}{3^2} = 2$.
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$\sin ^{-1}\left(\sum_{i=1}^{\infty} x^{i+1}-x \sum_{i=1}^{\infty}\left(\frac{x}{2}\right)^i\right)=\frac{\pi}{2}-\cos ^{-1}\left(\sum_{i=1}^{\infty}\left(-\frac{x}{2}\right)^i-\sum_{i=1}^{\infty}(-x)^i\right)$
lying in the interval $\left(-\frac{1}{2}, \frac{1}{2}\right)$ is. . . . .
(Here, the inverse trigonometric functions $\sin ^{-1} x$ and $\cos ^{-1} x$ assume values in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and $[0, \pi]$, respectively.)