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Find the sum to infinity of the series: 1 + 2/3 + 6/3² + 10/3³ + 14/3⁴ +... (the numerators form an arithmetic progression starting from the second term).

1. 4
2. 6
3. 2
4. 3

Correct answer: 3

Solution

Subtracting S/3 from S gives (2/3)S = 1 + 1/3 + 4*(sum of 1/9 + 1/27 +...) = 1 + 1/3 + (4/9)/(1-1/3) = 1 + 1/3 + 2/3 = 2, so S = 3.

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