The sum of the infinite series tan⁻¹(1/3) + tan⁻¹(1/7) + tan⁻¹(1/13) + ... is:
The sum of the infinite series tan⁻¹(1/3) + tan⁻¹(1/7) + tan⁻¹(1/13) + ... is:
- A. π/4
- B. π/2
- C. π
- D. 0
Answer: A) π/4
Explanation: The nth term is tan⁻¹(1 / (n² + n + 1)) = tan⁻¹((n+1 − n) / (1 + n(n+1))) = tan⁻¹(n+1) − tan⁻¹(n). Telescoping sum yields tan⁻¹(∞) − tan⁻¹(1) = π/2 − π/4 = π/4.
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