题目

MAT136H5 S 2025 - All Sections 5.2 preparation check

多重下拉选择题

Here are several questions related to the definition of an infinite series. a) For the series compute the th partial sum . [ Select ] -7 / 12 -47 / 60 5 / 7 5 / 12 -5 / 6 b) For the series compute the rd partial sum . [ Select ] 163 / 30 23 / 6 17 / 6 7 / 3 - 7 / 30 c) Which of the following statements are TRUE and which are FALSE? (In each case is the th partial sum of the series .) i) Every series must either converge or diverge, but never both. [ Select ] False True ii) If then the series converges [ Select ] True False iii) If then the series converges [ Select ] False True iv) If then the series diverges (=divergence of a series) [ Select ] False True v) If then [ Select ] False True

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思路分析

We begin by restating what the question is asking and listing the given options to make the task clear. Part a announces a series and asks to compute the nth partial sum, with the available selections for that partial sum being: -7/12, -47/60, 5/7, 5/12, -5/6. Since the actual series expression (the terms whose partial sums we are summing) is not included in the prompt we cannot explicitly show the algebra that leads from the series to a specific numeric partial sum. In a typical scenario, you would write down the first n terms of the series, sum them, and then simplify to obtain the partial sum in simplest terms, checking which option matches the simplification. The chosen option here is -7/12. If you had the concrete series, you would verify this by performing the finite sum and reducing the result to a single fraction, conf......Login to view full explanation

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The infinite series \[1-1/2+1/2-1/3+1/3-\ldots\] converges. What is its sum?

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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