Three series are defined below:

MATHEMATICS · Multi-part · A Level · Mathematics · Sequences and series · Sequences and series

Problem

Three series are defined below: S1=∑n=1∞n2n+3S_1=\displaystyle \sum_{n=1}^{\infty} \frac{n}{2n+3} S2=∑n=2∞1nln⁡nS_2=\displaystyle \sum_{n=2}^{\infty} \frac{1}{n\ln n} S3=∑n=1∞(−1)n+13n2S_3=\displaystyle \sum_{n=1}^{\infty} (-1)^{n+1}\frac{3}{n^{2}} (a) For each series, evaluate lim⁡n→∞an\displaystyle \lim_{n \to \infty} a_n, where ana_n is the nnth term. (b) Based solely on the limits found in part (a), determine which of the three series diverge. Provide justification for every conclusion. (c) For any series for which divergence could not be decided in part (b), state whether the limit test is inconclusive and explain the implication for the convergence of that series.

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