Question

If \( |a| < 1 \) and \( |b| < 1 \), then the sum of the series \[ a(a + b) + a^2(a^2 + b^2) + a^3(a^3 + b^3) + \cdots \]

• A. \( \frac{a}{1 - ab} \)
• B. \( \frac{a^2}{1 - ab} \)
• C. \( \frac{b^2}{1 - b^2} \)
• D. \( \frac{a}{1 - b^2} \)

Hint:

When summing series involving powers of \(a\) and \(b\), identify the pattern and use geometric series formulas if applicable.

Answer 1

The Correct Option is B

The sum of the series is given by: \[ S = a(a + b) + a^2(a^2 + b^2) + a^3(a^3 + b^3) + \cdots \] Factor out \( a \) from each term to get: \[ S = a \left[ (a + b) + a(a^2 + b^2) + a^2(a^3 + b^3) + \cdots \right]. \] This series can be simplified and summed, leading to the result: \[ S = \frac{a^2}{1 - ab}. \]