Evaluate the sum of the geometric series ∑_{n=0}^∞ (1/3)^n.

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Multiple Choice

Evaluate the sum of the geometric series ∑_{n=0}^∞ (1/3)^n.

When summing an infinite geometric series, the total exists and equals the first term divided by one minus the common ratio, provided the ratio's absolute value is less than 1. Here the first term is 1 and the ratio is 1/3, so the sum is 1/(1 − 1/3) = 1/(2/3) = 3/2. The terms shrink by a factor of three each time, so the tail contributes a finite amount and the partial sums approach 3/2. The other numbers would only arise from a different first term or ratio.

When summing an infinite geometric series, the total exists and equals the first term divided by one minus the common ratio, provided the ratio's absolute value is less than 1. Here the first term is 1 and the ratio is 1/3, so the sum is 1/(1 − 1/3) = 1/(2/3) = 3/2. The terms shrink by a factor of three each time, so the tail contributes a finite amount and the partial sums approach 3/2. The other numbers would only arise from a different first term or ratio.

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