题目
题目

MAT137Y1 LEC 20249: Calculus with Proofs (all lecture sections) Pre-Class Quiz 53 (11.7 and 11.8)

多项选择题

Let  { 𝑎 𝑛 } 𝑛 = 0 ∞ , { 𝑏 𝑛 } 𝑛 = 0 ∞   and  { 𝑐 𝑛 } 𝑛 = 0 ∞   be positive sequences that are divergent to ∞ . Assume that 𝑎 𝑛 << 𝑏 𝑛 . Which of the following statements must be true? Select all the correct answers.

选项
A.IF ∀ 𝑛 ∈ 𝑁 ,     𝑏 𝑛 = 2 𝑐 𝑛 ,  THEN 𝑎 𝑛 << 𝑐 𝑛 .
B.IF 𝑏 𝑛 << 𝑐 𝑛 ,  THEN 𝑎 𝑛 << 𝑐 𝑛   .
C.IF lim 𝑛 → ∞ 𝑐 𝑛 𝑏 𝑛 = 0 ,  THEN lim 𝑛 → ∞ 𝑎 𝑛 𝑐 𝑛 = 0 .
D.lim 𝑛 → ∞ 𝑏 𝑛 𝑎 𝑛  DNE
E.IF ∀ 𝑛 ∈ 𝑁 ,     𝑐 𝑛 = 𝑎 𝑛 + 𝑏 𝑛 2   ,  THEN 𝑎 𝑛 << 𝑐 𝑛 .
F.IF ∀ 𝑛 ∈ 𝑁 ,   𝑐 𝑛 = min { 𝑎 𝑛 , 𝑏 𝑛 }   ,  THEN ∀ 𝑛 ∈ 𝑁 ,   𝑐 𝑛 = 𝑎 𝑛  .
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思路分析
We are given three positive sequences a_n, b_n, c_n that diverge to infinity, with a_n = o(b_n) (i.e., a_n << b_n). We examine each option to determine whether it must be true under these assumptions. Option 1: IF ∀n, b_n = 2 c_n, THEN a_n << c_n. Since b_n = 2 c_n, we have a_n / c_n = (a_n / b_n) * (b_n / c_n) = (a_n / b_n) * 2. Because a_n / b_n → 0 by a_n << b_n, multiplying by the constant 2 preserves the limit, so a_n / c_n → 0. Th......Login to view full explanation

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类似问题

Let  { 𝑎 𝑛 } 𝑛 = 0 ∞ ,   { 𝑏 𝑛 } 𝑛 = 0 ∞   and   { 𝑅 𝑛 } 𝑛 = 0 ∞  be POSITIVE sequences that are DIVERGENT TO ∞ . Assume that 𝑎 𝑛 << 𝑅 𝑛   and   𝑏 𝑛 << 𝑅 𝑛   . Which of the following statements must be true? Select all the correct answers.

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