Single correct option
Find the sum of the infinite series 1 + 2/3 + 3/9 + 4/27 + 5/81 + …
- A. 3/2
- B. 9/4
- C. 2
- D. 8/3
Show answer and solution
Answer: (B) 9/4
Let S = 1 + 2/3 + 3/9 + 4/27 + …
Then S/3 = 1/3 + 2/9 + 3/27 + …
Subtracting: S − S/3 = 1 + 1/3 + 1/9 + … = 1/(1 − 1/3) = 3/2.
So (2/3)S = 3/2 → S = 9/4.
Formula: Σ n·rⁿ⁻¹ = 1/(1 − r)² for |r| < 1; with r = 1/3 this is 1/(4/9) = 9/4.
Then S/3 = 1/3 + 2/9 + 3/27 + …
Subtracting: S − S/3 = 1 + 1/3 + 1/9 + … = 1/(1 − 1/3) = 3/2.
So (2/3)S = 3/2 → S = 9/4.
Formula: Σ n·rⁿ⁻¹ = 1/(1 − r)² for |r| < 1; with r = 1/3 this is 1/(4/9) = 9/4.
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