Consider the series with terms

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Consider the series ∑n=1∞an(k)\displaystyle \sum_{n=1}^{\infty} a_n(k) with terms an(k)=knna_n(k)=\dfrac{k^{n}}{n}, where kk is a real parameter. (a) Evaluate L(k)=lim⁡n→∞an(k)L(k)=\displaystyle \lim_{n\to\infty} a_n(k) in terms of kk. Clearly state all cases that arise. (b) Using your result from part (a), identify all real numbers kk for which the series cannot converge. Provide a complete justification based on the behavior of the sequence {an(k)}\{a_n(k)\}. (c) When ∣k∣≤1|k|\le 1, does the information obtained from part (a) guarantee convergence or divergence of the series? Explain fully, citing the relevant reasoning about the limit you computed.

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