Question

\[ \sum_{r=1}^{\infty} \left(3\cdot 2^{-r} - 2\cdot 3^{1-r}\right) \]

• A. 2
• B. 1/2
• C. 1
• D. 0

Hint:

Break series into standard GP form before summing.

Answer 1

The Correct Option is D

Concept: Split into two GP series.

Step 1: Evaluate first sum.
\[ \sum 3\cdot 2^{-r} = 3\left(\frac{1/2}{1-1/2}\right) = 3 \]

Step 2: Second sum.
\[ \sum 2\cdot 3^{1-r} = 2\cdot \frac{3}{2} = 3 \]

Step 3: Final.
\[ 3 - 3 = 0 \]