Consider the telescoping series ∑_{n=1}^{∞} (1/n − 1/(n+1)). Does it converge, and if so, to what value?

Consider the telescoping series ∑_{n=1}^{∞} (1/n − 1/(n+1)). Does it converge, and if so, to what value?

• A. Yes, converges to 1/2
• B. Yes, converges to 0
• C. No, diverges
• D. Yes, converges to 1/2

Answer: (D)

The key idea is telescoping. When you sum from n=1 to N, the middle terms cancel: (1 − 1/2) + (1/2 − 1/3) + … + (1/N − 1/(N+1)) leaves 1 − 1/(N+1). As N grows, 1/(N+1) goes to 0, so the partial sums approach 1. Therefore the series converges, and its sum is 1.

The key idea is telescoping. When you sum from n=1 to N, the middle terms cancel: (1 − 1/2) + (1/2 − 1/3) + … + (1/N − 1/(N+1)) leaves 1 − 1/(N+1). As N grows, 1/(N+1) goes to 0, so the partial sums approach 1. Therefore the series converges, and its sum is 1.