Evaluate the sum: (1/1 − 1/2) + (1/2 − 1/3) + (1/3 − 1/4) + … up to n terms.

Given data

• Telescoping series: (1/1 − 1/2) + (1/2 − 1/3) + … for n terms.

Concept / Approach

• Write out first few terms to see cancellation (telescoping).

Step-by-step calculation

S_n = (1 − 1/2) + (1/2 − 1/3) + (1/3 − 1/4) + … + (1/n − 1/(n + 1))All intermediate terms cancel, leaving S_n = 1 − 1/(n + 1)Therefore, S_n = 1 − 1/(n + 1)

Verification

For n = 3: (1 − 1/2) + (1/2 − 1/3) + (1/3 − 1/4) = 1 − 1/4 = 3/4.

Common pitfalls

• Stopping at 1 − 1/n; the final term is 1/(n + 1), not 1/n.

Final Answer

1 − 1/(n + 1)