Question

The sum of the series \( 1 - \frac{2}{3} + \frac{2.4}{3.6} - \frac{2.4.6}{3.6.9} + \cdots \infty \) is:

• A. \( \frac{3}{5} \)
• B. \( \left( \frac{2}{5} \right)^{2/3} \)
• C. \( \frac{2}{5} \)
• D. \( \left( \frac{3}{5} \right)^{2/3} \)

Hint:

For infinite geometric series, use the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio.

Answer 1

The Correct Option is A

The given series is a type of infinite geometric series. The general form of the series is: \( S = 1 - \frac{2}{3} + \frac{2.4}{3.6} - \frac{2.4.6}{3.6.9} + \cdots \) 

Step 1: Express this as a geometric series with first term \( 1 \) and common ratio \( \frac{-2}{3} \). The sum of an infinite geometric series is given by: \( S = \frac{a}{1 - r} \) Where \( a \) is the first term and \( r \) is the common ratio. Here, \( a = 1 \) and \( r = -\frac{2}{3} \). \( S = \frac{1}{1 - \left(-\frac{2}{3}\right)} = \frac{1}{1 + \frac{2}{3}} = \frac{1}{\frac{5}{3}} = \frac{3}{5} \)