The sum of the geometric series ∑_{n=0}^∞ (1/2)^n equals how much?

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Multiple Choice

The sum of the geometric series ∑_{n=0}^∞ (1/2)^n equals how much?

Explanation:
Infinite geometric series converge to a finite value when the common ratio has absolute value less than 1, and the sum is the first term divided by 1 minus the ratio. Here the first term is 1 (since (1/2)^0 = 1) and the common ratio is 1/2. So the sum is 1/(1 - 1/2) = 1/(1/2) = 2. You can also see this from the partial sums S_N = a(1 - r^{N+1})/(1 - r) = 2(1 - (1/2)^{N+1}) → 2 as N → ∞. Therefore the infinite sum equals 2.

Infinite geometric series converge to a finite value when the common ratio has absolute value less than 1, and the sum is the first term divided by 1 minus the ratio. Here the first term is 1 (since (1/2)^0 = 1) and the common ratio is 1/2. So the sum is 1/(1 - 1/2) = 1/(1/2) = 2. You can also see this from the partial sums S_N = a(1 - r^{N+1})/(1 - r) = 2(1 - (1/2)^{N+1}) → 2 as N → ∞. Therefore the infinite sum equals 2.

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