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MAT136H5 S 2025 - All Sections 5.4 preparation check

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Read Theorem 5.12 in the textbook Links to an external site. . Suppose 𝑎 𝑛 ≥ 0 for all 𝑛 ≥ 1 and 𝑏 𝑛 ≥ 0 for all 𝑛 ≥ 1 . a) If ∑ 𝑛 = 1 ∞ 𝑏 𝑛 converges and lim 𝑛 → ∞ 𝑎 𝑛 𝑏 𝑛 = 5 then ∑ 𝑛 = 1 ∞ 𝑎 𝑛 [ Select ] might converge or diverge (there is not enough information) converges diverges b) If ∑ 𝑛 = 1 ∞ 𝑏 𝑛 converges and lim 𝑛 → ∞ 𝑎 𝑛 𝑏 𝑛 = 0 then ∑ 𝑛 = 1 ∞ 𝑎 𝑛 [ Select ] diverges converges might converge or diverge (there is not enough information) c) If ∑ 𝑛 = 1 ∞ 𝑏 𝑛 diverges and lim 𝑛 → ∞ 𝑎 𝑛 𝑏 𝑛 = 0 then ∑ 𝑛 = 1 ∞ 𝑎 𝑛 [ Select ] might converge or diverge (there is not enough information) diverges converges d) If ∑ 𝑛 = 1 ∞ 𝑏 𝑛 diverges and lim 𝑛 → ∞ 𝑎 𝑛 𝑏 𝑛 = 200 then ∑ 𝑛 = 1 ∞ 𝑎 𝑛 diverges

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We are given nonnegative sequences a_n and b_n with a_n ≥ 0 and b_n ≥ 0 for all n ≥ 1, and relationships between the series ∑ b_n and the limit of a_n b_n. We will examine each option carefully. a) If ∑_{n=1}^∞ b_n converges and lim_{n→∞} a_n b_n = 5, then ∑_{n=1}^∞ a_n [Select]. - Since ∑ b_n converges, we know b_n → 0 as n → ∞. The condition lim a_n b_n = 5 implies that for large n, a_n behaves roughly like 5 / b_n, which forces a_n to grow without bound because b_n → 0. In particular, a_n does not approach 0; instead......Login to view full explanation

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