Evaluate the telescoping sum: sum_{k=1}^n (1 / (k*(k + 1))).
Answer and explanation
Correct answer: D. n / (n + 1)
- Step 1: Decompose the summand into partial fractions: 1 / (k*(k + 1)) = 1/k - 1/(k + 1).
- Step 2: Write out terms: (1/1 - 1/2) + (1/2 - 1/3) + ... + (1/n - 1/(n + 1)).
- Step 3: All intermediate terms cancel, leaving 1 - 1/(n + 1) = n / (n + 1).
Answer
n / (n + 1).
Why not the others: 1 / (n + 1) is the remaining subtracted term, not the total sum; (n + 1) / n is the reciprocal; 1 / n is an approximation.
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