题目
题目

MAT136H5 S 2025 - All Sections 5.2 preparation check

多重下拉选择题

Suppose that and are convergent series and that   and . Also . Find the sum of each of the following series, or determine that it is divergent, or that not enough information is given. Use Theorem 5.7 as appropriate.  a)       [ Select ] 2 5 3 Series is divergent Not enough information 8 b)     [ Select ] -1 Not enough information 8 -2 -9 Series is divergent c)       [ Select ] 9 Not enough information Series is divergent 6 1 d)       [ Select ] Series is divergent 2 3 1 Not enough information

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思路分析
The problem statement as provided is partially garbled, and the exact definitions of the series in parts (a)–(d) are not visible here. To give meaningful guidance, I will reason in a way that mirrors how a student would approach such a question, while clearly noting where information is missing and how that affects the conclusions. In each part, I will evaluate the listed options and explain why each could be correct or incorrect under typical interpretations of Theorem 5.7 (which often concerns products or convolutions of convergent series) and the usual convergence tests. Remember, without the explicit series terms or the exact form of Theorem 5.7 as presented in your course, some judgments necessarily hinge on standard interpretations of these topics. General background (useful for all parts): - If two series ∑a_n and ∑b_n are convergent, their Cauchy product ∑c_n (where c_n = sum_{k=0}^n a_k b_{n-k}) may converge to the product (∑a_n)(∑b_n) under certain conditions. A common sufficient condition is that at least one of the two series is absolutely convergent. If neither is absolutely convergent, the Cauchy product need not converge, and its sum may differ from (∑a_n)(∑b_n) or be undefined. - The statement ‘Not enough information’ is often the correct choice when the problem does not provide details about absolute convergence, the exact definitions of the series, or the precise form of the ope......Login to view full explanation

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Question text(1) Here is a convergent infinite series 1+1/2+1/4+1/8+1/16+...What kind of infinite series are we dealing with?Answer 1 Question 16[select: , arithmetic, geometric, p-series, harmonic, telescoping, none of the above]What is the fifth partial sum of this series (written in lowest terms)? Answer 2 Question 16[input] What's its sum? Answer 3 Question 16[input] (2). Here is another convergent infinite series 1+1/4+1/9+1/16+1/25+... What kind of infinite series are we dealing with?Answer 4 Question 16[select: , arithmetic, geometric, p-series, harmonic, telescoping, none of the above]What is the third partial sum of this series? Answer 5 Question 16[input] What is the integer part of its sum? Answer 6 Question 16[input] (3) Here is yet another converging infinite series What kind of infinite series are we dealing with?Answer 7 Question 16[select: , arithmetic, geometric, p-series, harmonic, telescoping, none of the above]What is the sum of the first three terms of this series? Answer 8 Question 16[input] What's its sum? Answer 9 Question 16[input] Check Question 16

Consider the series ∑ 𝑛 = 1 ∞ 0.01 . The terms are 𝑎 𝑛 = 0.01 .   a) Find the following partial sums: 𝑆 1 = [ Select ] 0.01 0.02 1 0 0.03 𝑆 2 =   [ Select ] 0.02 0.04 2 1 0.01 𝑆 3 = [ Select ] 3 4 0.03 0 0.01 𝑆 4 = [ Select ] 0.4 0.04 4 1 0.01   b) Find the limits:     lim 𝑘 → ∞ 𝑆 𝑘 =   [ Select ] infinity 0.04 0.01 negative infinity 0 and lim 𝑛 → ∞ 𝑎 𝑛 =   [ Select ] negative infinity infinity 0.01 1 0   c) Does the series ∑ 𝑛 = 1 ∞ 0.01  converge or diverge? [ Select ] The series converges There is not enough information to tell The series diverges   d) Suppose another series ∑ 𝑛 = 1 ∞ 𝑏 𝑛  has some unknown terms 𝑏 𝑛   but we know that lim 𝑛 → ∞ 𝑏 𝑛 = 0.01   (this means the numbers 𝑏 𝑛 are close to 0.01, but not necessarily equal to 0.01.)  What can be said about the convergence of the series ∑ 𝑛 = 1 ∞ 𝑏 𝑛  ? [ Select ] The series converges The series diverges There is not enough information to tell

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