STD 11 - 8. sequence and series — JEE कक्षा 11 साइन्स — Question

$ = 0.4 + 0.023 + 0.00023 + 0.0000023 + ........\infty $

$ = \frac{4}{{10}} + \frac{{23}}{{{{10}^3}}} + \frac{{23}}{{{{10}^5}}} + \frac{{23}}{{{{10}^7}}}........\infty $

$ = \frac{4}{{10}} + \frac{{23}}{{{{10}^3}}}\left[ {1 + \frac{1}{{{{10}^2}}} + \frac{1}{{{{10}^4}}} + ..........\infty } \right]$

$ = \frac{4}{{10}} + \frac{{23}}{{1000}}\left( {\frac{1}{{1 - \frac{1}{{{{10}^2}}}}}} \right) = \frac{4}{{10}} + \frac{{23}}{{990}} = \frac{{419}}{{990}}$.