Mathematics

Problem

Consider the series

\displaystyle \sum_{n=1}^{\infty} (-1)^{n}\dfrac{n}{n+1}

. Which statement correctly identifies the failed hypothesis of the alternating‐series convergence criteria and the resulting conclusion about the series? a) The monotone‐decreasing hypothesis fails, so the series diverges. b) The limit hypothesis fails, but because the series alternates, the

n

th–term divergence test is inconclusive. c) The limit hypothesis fails, so the series diverges by the

n

th–term test. d) Neither hypothesis fails, so the series converges.

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