Define by b_n= cases 1 n , & n odd , 2 n , & n even

MATHEMATICS · Subjective · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Define bnb_n by bn={1n,n odd,2n,n even.b_n=\begin{cases}\dfrac{1}{n}, & n\text{ odd},\\ \dfrac{2}{n}, & n\text{ even}.\end{cases} Examine the alternating series ∑n=1∞(−1)n−1bn\displaystyle \sum_{n=1}^{\infty}(-1)^{n-1}b_n. Does the sequence {bn}\{b_n\} satisfy the requirement that its terms be eventually non-increasing? Explain your reasoning and state what, if anything, can be concluded about convergence from the usual alternating-series conditions alone.

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