Question

$\frac{2}{1}.\frac{1}{3} + \frac{3}{2}.\frac{1}{9} + \frac{4}{3}.\frac{1}{27} + \frac{5}{4}.\frac{1}{81} + ......\infty =$

• A. $\frac{1}{2} - \log_{e}\frac{2}{3}$
• B. $- \log_{e}\frac{2}{3}$
• C. $\frac{1}{2} + \log_{e}\left( \frac{2}{3} \right)$
• D. None of these

Answer 1

Answer: (A)

$\frac{1}{2} - \log_{e}\frac{2}{3}$

Answer 2

$n^{- r}$,

Where $\log_{10}\left( \frac{n}{n - 1} \right)$

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

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

Trick : As the sum of the series upto 3 or 4 terms is approximately 0.9. Obviously (1) gives the value nearer to 0.9.