Compute the value of the telescoping series ∑_{n=1}^∞ (1/n - 1/(n+1)).

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Multiple Choice

Compute the value of the telescoping series ∑_{n=1}^∞ (1/n - 1/(n+1)).

This is a telescoping series. When you form the partial sum S_N = sum from n=1 to N of (1/n - 1/(n+1)), the terms cancel in sequence: (1 - 1/2) + (1/2 - 1/3) + ... + (1/N - 1/(N+1)) leaves just 1 - 1/(N+1). So S_N = 1 - 1/(N+1). As N grows, 1/(N+1) → 0, hence the sum to infinity is 1. The option that equals 1 is the correct choice.

This is a telescoping series. When you form the partial sum S_N = sum from n=1 to N of (1/n - 1/(n+1)), the terms cancel in sequence: (1 - 1/2) + (1/2 - 1/3) + ... + (1/N - 1/(N+1)) leaves just 1 - 1/(N+1). So S_N = 1 - 1/(N+1). As N grows, 1/(N+1) → 0, hence the sum to infinity is 1. The option that equals 1 is the correct choice.

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