1 - frac{2}{3} + frac{2 cdot 4}{3 cdot 6} - f | vedclass

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Question

(A) The given series is $1 - \frac{2}{3} + \frac{2 \cdot 4}{3 \cdot 6} - \frac{2 \cdot 4 \cdot 6}{3 \cdot 6 \cdot 9} + \ldots \infty$.This can be rewr

Vedclass · Accepted Answer

$\frac{3}{5}$

Vedclass · Answer

(A) The given series is $1 - \frac{2}{3} + \frac{2 \cdot 4}{3 \cdot 6} - \frac{2 \cdot 4 \cdot 6}{3 \cdot 6 \cdot 9} + \ldots \infty$.This can be rewritten as $1 - \frac{2}{3} + \left(\frac{2}{3}\right)^2 - \left(\frac{2}{3}\right)^3 + \ldots \infty$.This is an infinite geometric series with first term $a = 1$ and common ratio $r = -\frac{2}{3}$.The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$.Substituting the values,$S = \frac{1}{1 - (-\frac{2}{3})} = \frac{1}{1 + \frac{2}{3}} = \frac{1}{\frac{5}{3}} = \frac{3}{5}$.

$1 - \frac{2}{3} + \frac{2 \cdot 4}{3 \cdot 6} - \frac{2 \cdot 4 \cdot 6}{3 \cdot 6 \cdot 9} + \ldots \infty =$

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