题目
题目

MTH1030 -1035 - S1 2025 MTH1030/5 Week 8 lesson quiz: Infinite sequences and series

简答题

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

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思路分析
We are given the infinite series: 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + ... A helpful way to analyze it is to look at how the terms cancel in pairs starting from the second term. - The first term stands alone: 1.......Login to view full explanation

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Given the series    has infinite terms, what sum will this series approach but never reach? (type number only in box)

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

Consider the series ∑ 𝑛 = 1 ∞ 0.01 . The terms are 𝑎 𝑛 = 0.01 .   a) Find the following partial sums: 𝑆 1 = [ Select ] 1 0.03 0.02 0.01 0 𝑆 2 =   0.02 𝑆 3 = [ Select ] 0.03 4 0.01 0 3 𝑆 4 = [ Select ] 0.4 0.01 0.04 4 1   b) Find the limits:     lim 𝑘 → ∞ 𝑆 𝑘 =   [ Select ] negative infinity 0 infinity 0.01 0.04 and lim 𝑛 → ∞ 𝑎 𝑛 =   [ Select ] 1 infinity 0.01 negative infinity 0   c) Does the series ∑ 𝑛 = 1 ∞ 0.01  converge or diverge? 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  ? The series diverges

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