Consider the series _ n=1 ^ 2n n+1

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Consider the series ∑n=1∞2nn+1.\sum_{n=1}^{\infty} \frac{2n}{n+1}. Which statement provides the most direct justification for its convergence or divergence? (a) It converges absolutely by the ratio test. (b) It converges by the alternating series test because the terms decrease and alternate in sign. (c) It diverges because lim⁡n→∞2nn+1≠0\displaystyle\lim_{n\to\infty}\frac{2n}{n+1}\neq 0. (d) It converges conditionally by comparison with the harmonic series since 2nn+1<1n\dfrac{2n}{n+1}<\dfrac{1}{n}.

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