Mathematics

Mathematics

Problem

For the alternating series ∑n=1∞(−1)nbn\displaystyle \sum_{n=1}^{\infty} (-1)^{n} b_n with bn>0b_n>0, which of the following sets of conditions is sufficient to guarantee that the series converges? a) lim⁡n→∞bn=0 and bn+1≤bn for all n\displaystyle \lim_{n\to\infty} b_n=0 \text{ and } b_{n+1}\le b_n \text{ for all } n b) bn is bounded and bn+1≤bn for all nb_n \text{ is bounded and } b_{n+1}\le b_n \text{ for all } n c) lim⁡n→∞bn=L (finite) and bn+1≤bn for all n\displaystyle \lim_{n\to\infty} b_n=L \text{ (finite) and } b_{n+1}\le b_n \text{ for all } n d) lim⁡n→∞bn=0\displaystyle \lim_{n\to\infty} b_n=0 only

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