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Question
If |x| < 1 and y = 1 + x + x2 + x3 + ..., then write the value of \[\frac{dy}{dx}\]
Solution
The given series is a geometric series where a = 1 and r = x.
\[f\left( x \right) = 1 + x + x^2 + x^3 + . . . = \frac{1}{1 - x}\]
\[\left( \text{ Sum of the infinite series of a geometric series is }\frac{a}{1 - r}. \right)\]
\[f'\left( x \right) = \frac{- 1}{(1 - x )^2}\frac{d}{dx}(1 - x)\]
\[ = \frac{- 1}{(1 - x )^2}( - 1)\]
\[ = \frac{1}{(1 - x )^2}\]
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