Problem 1. The Alternating Series Test states that if the positive sequence {b„} is (1) decreasing, and (2) convergent \n+1 to 0, then the series >(-1)n+'bn converges. But what if we drop the assumption that {bn} is decreasing? Is the n=1 result still true? Consider the series defined by 8. 1 1 + 16 1 1 1 - + 4 1 -- + 3 1 1 E(-1)"+1bn; - - .. 8. 4 n 2n n=1 where the sequence b, is defined by 1 1 1 1 1 1 1 1 1 1 2'4'3'8'4' 16'5' 32 2n n (a) Does this sequence {b„} satisfy the assumptions of the Alternating Series Test? Which does it satisfy, and which does it fail? 1 Σ E(;-). Suppose that it did (b) Show that this series diverges. (Hint: This series can also be written as 2n n=1 converge. If you add a certain geometric series to it, you're adding two convergent series together, so you should get another convergent series-but do you?) (c) Is the Alternating Series Test wrong? Explain why not.

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