题目
题目
多项填空题

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

题目图片
查看解析

查看解析

标准答案
Please login to view
思路分析
We are given three parts in this composite question about different infinite series, each with its own type and sums. I will go through each option and explain why it is or isn’t correct for the described series in that part, using clear reasoning and pointing out common pitfalls. Part (1): First series 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... Option: arithmetic Why it’s not correct: An arithmetic series has constant difference between consecutive terms. Here the differences are 1/2, 1/4, 1/8, ... which are not constant. So arithmetic is incorrect. Option: geometric Why it is correct: Each term is multiplied by a constant ratio of 1/2 to get the next term (1 -> 1/2 -> 1/4 -> 1/8 -> ...). This is the hallmark of a geometric series. Option: p-series Why it’s not correct: p-series have terms of the form 1/n^p with n rising through integers. Our terms are powers of 1/2, not 1/n^p, so not a p-series. Op......Login to view full explanation

登录即可查看完整答案

我们收录了全球超50000道考试原题与详细解析,现在登录,立即获得答案。

类似问题

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

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

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

更多留学生实用工具

加入我们，立即解锁 海量真题 与 独家解析，让复习快人一步！