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  4. The value of 1+(1/32)+(1.4/1.2) (1/34)+(1.4 .7/1.2 .3) (1/36)+ ldots is

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Q. The value of $1+\frac{1}{3^{2}}+\frac{1.4}{1.2} \frac{1}{3^{4}}+\frac{1.4 .7}{1.2 .3} \frac{1}{3^{6}}+\ldots$ is

Binomial Theorem

Solution:

Let $S=1+\frac{1}{3^{2}}+\frac{1.4}{1.2} \frac{1}{3^{4}}+\frac{1.4 .7}{1.2 .3} \frac{1}{3^{6}}+\ldots$
$\Rightarrow S=1+\left(\frac{1}{3}\right)\left(\frac{1}{3}\right)+\frac{\frac{1}{3} \cdot \frac{4}{3}}{1.2}\left(\frac{1}{3}\right)^{2}+\frac{\frac{1}{3} \cdot \frac{4}{3} \cdot \frac{7}{3}}{1.2 .3}\left(\frac{1}{3}\right)^{3}+\ldots$
$ \Rightarrow S=1 +\left(\frac{1}{3}\right)\left(\frac{1}{3}\right)+\frac{\frac{1}{3}\left(1+\frac{1}{3}\right)}{2 !}\left(\frac{1}{3}\right)^{2} $
$+ \frac{\frac{1}{3}\left(1+\frac{1}{3}\right)\left(2+\frac{1}{3}\right)}{3 !}\left(\frac{1}{3}\right)^{3}+\ldots $
$=\left(1-\frac{1}{3}\right)^{-\frac{1}{3}}=\left(\frac{2}{3}\right)^{-\frac{1}{3}}=\left(\frac{3}{2}\right)^{\frac{1}{3}} $