For each real number

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

Problem

For each real number α\alpha, define the series S(α)=∑n=1∞nαnα+1S(\alpha)=\displaystyle \sum_{n=1}^{\infty} \dfrac{n^{\alpha}}{n^{\alpha}+1}. (a) Determine L(α)=lim⁡n→∞nαnα+1L(\alpha)=\displaystyle \lim_{n\to\infty} \dfrac{n^{\alpha}}{n^{\alpha}+1}. (b) Using the value of L(α)L(\alpha), find all real α\alpha for which the series S(α)S(\alpha) diverges. Justify your answer rigorously. (c) Consider the alternating series ∑n=1∞(−1)nnαnα+1\displaystyle \sum_{n=1}^{\infty} (-1)^n \dfrac{n^{\alpha}}{n^{\alpha}+1}. For which real α\alpha does this alternating series diverge? Provide full reasoning, explicitly referencing the behavior of the underlying sequence of terms.

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