Which statement best describes the convergence of the series ?

MATHEMATICS · Single correct · IB Diploma Programme · Mathematics: Analysis and Approaches · Number and Algebra · Number and Algebra

Problem

Which statement best describes the convergence of the series ∑n=0∞(−1)n (n+1)!(n+2)!\displaystyle \sum_{n=0}^{\infty} (-1)^{n}\,\frac{(n+1)!}{(n+2)!}? (a) It converges conditionally; the alternating series converges, but the series of absolute values diverges. (b) It converges absolutely; applying the ratio test gives a limit of 00, so the series converges. (c) It diverges; simplifying the factorials gives the harmonic series, which is known to diverge. (d) It converges to exactly 11 because the factorial terms create a telescoping pattern.

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