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

Question

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.

BlackTom AI · Accepted Answer

Let us restate the problem to ensure clarity: we have positive sequences a_n, b_n, and R_n, each defined for n from 0 to infinity, with R_n diverging to infinity. We are given that a_n << R_n and b_n << R_n, meaning a_n / R_n -> 0 and b_n / R_n -> 0. We are asked to determine which of the following statements must be true: Option 1: a_n b_n << R_n^2 Option 2: a_n + b_n << 2 R_n Now we examine each option step by step. Option 1: a_n b_n << R_n^2 - Start by considering the ratio (a_n b_n) / R_n^2. By the given relations, a_n / R_n -> 0 and b_n / R_n -> 0. - Multiply these two limits: (a_n / R_n) * (b_n / R_n) -> 0 * 0 = 0. This product is exactly (a_n b_n) / R_n^2. - Therefore (a_n b_n) / R_n^2 -> 0, which means a_n b_n = o(R_n^2). This matches the meaning of the notation "<< R_n^2" in the context of little-o notation. - Conclusion for Option 1: The statement must be true, since the product of two quantities each negligible compared to R_n is negligible compared to R_n^2. Option 2: a_n + b_n << 2 R_n - Consider the ratio (a_n + b_n) / (2 R_n). By linearity, this equals (a_n / (2 R_n)) + (b_n / (2 R_n)). - Since a_n / R_n -> 0, we also have a_n / (2 R_n) -> 0. Similarly, b_n / R_n -> 0 implies b_n / (2 R_n) -> 0. - The sum of two quantities each tending to 0 also tends to 0, so (a_n + b_n) / (2 R_n) -> 0. - This means a_n + b_n = o(R_n). In the given notation, it is represented as a_n + b_n << 2 R_n. - Conclusion for Option 2: The statement must be true, since the sum of two quantities each negligible relative to R_n is itself negligible relative to R_n (and hence relative to 2 R_n as well). In summary, both options rely on the foundational fact that a_n and b_n are each little-o of R_n; their product is thus little-o of R_n^2, and their sum is little-o of R_n, which corresponds to the provided comparisons. The logical steps above show how the little-o relations propagate through multiplication and addition, yielding the conclusions for both statements.

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

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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类似问题

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.

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