In this exercise, we examine one of the conditions of the Alternating Series Test. Consider the...

In this exercise, we examine one of the conditions of the Alternating Series Test. Consider the alternating series

1−1+1/2−1/4+1/3−1/9+1/4−1/16+⋯,

where the terms are selected alternately from the sequences {1/n} and {−1/n^2}.

Explain why the nth term of the given series converges to 0 as n goes to infinity.
Rewrite the given series by grouping terms in the following manner:

(1−1)+(1/2−1/4)+(1/3−1/9)+(1/4−1/16)+⋯.

Use this regrouping to determine if the series converges or diverges.
Explain why the condition that the sequence {an}{an} decreases to a limit of 0 is included in the Alternating Series Test.

Expert Solution

milcah answered 2 years ago

Alternating Series Test. Let (an) be a sequence satisfying (i) a1 ≥ a2 ≥ a3 ≥...

Alternating Series Test. Let (an) be a sequence satisfying (i) a1 ≥ a2 ≥ a3 ≥ · · · ≥ an ≥ an+1 ≥ · · · and (ii) (an) → 0. Show that then the alternating series X∞ n=1 (−1)n+1an converges using the following two different approaches. (a) Show that the sequence (sn) of partial sums, sn = a1 − a2 + a3 − · · · ± an is a Cauchy sequence Alternating Series Test. Let (an) be...

Prove series 2, (-1/2), (2/9), (-1/8) is convergent by the alternating series test and find the...

Prove series 2, (-1/2), (2/9), (-1/8) is convergent by the alternating series test and find the number of terms required to estimate the sum of the series with an error of less than 0.05

Introduction Through a series of activities we will examine some of the principles of genetics and...

Introduction Through a series of activities we will examine some of the principles of genetics and techniques developed by geneticists to predict mating outcomes and understand how genetic information is passed from generation to generation. Before coming to lab, refer to your textbook or other references and write definitions for the following words on a separate sheet of paper: chromosome, genes, locus, allele, dominant allele, recessive allele, genotype, phenotype, gamete, haploid, diploid, monohybrid, dihybrid, homozygous, heterozygous, linked genes, autosomal chromosomes,...

In this exercise, we examine the effect of the margin of error on determining the sample...

In this exercise, we examine the effect of the margin of error on determining the sample size needed. Find the sample size needed to give, with 99% confidence, a margin of error within ±7. Within ±3. Within ±1. Assume that we use σ˜=20 as our estimate of the standard deviation in each case. Round your answers up to the nearest integer. ME=7 : n= ? ME=3 : n= ? ME=1 : n= ?

CHAPTER 2-4 Exercise 2.24 In this exercise, we examine the effect of combining investments with positively...

CHAPTER 2-4 Exercise 2.24 In this exercise, we examine the effect of combining investments with positively correlated risks, negatively correlated risks, and uncorrelated risks. A firm is considering a portfolio of assets. The portfolio is comprised of two assets, which we will call ''A" and "B." Let X denote the annual rate of return from asset A in the following year, and let Y denote the annual rate of return from asset B in the following year. Suppose that E(X)...

1. In this exercise, we examine the effect of the value of the estimated standard deviation...

1. In this exercise, we examine the effect of the value of the estimated standard deviation on determining the sample size needed. Find the sample size needed to give, with 90% confidence, a margin of error within ±4, if the estimated standard deviation is σ˜=60. If the estimated standard deviation is σ˜=20. If the estimated standard deviation is σ˜=10. σ˜=60 : n=? σ˜=20 : n= ? σ˜=10 : n=? 2. In this exercise, we examine the effect of the confidence...

In this exercise, the first of a series, we will make connections between the physics you...

In this exercise, the first of a series, we will make connections between the physics you have been learning (in this case, kinematics) and how it is used by people in their work and research. Today we consider an example from kinesiology research, based on a 2000 paper from the Journal of Measurement in Physical Education and Exercise Science (nota bene: you do not have to read this paper, or any of the hyperlinks to follow the exercise – it...

Use the Alternating Series Estimation Theorem to estimate the range of values of x for which...

Use the Alternating Series Estimation Theorem to estimate the range of values of x for which the given approximation is accurate to within the stated error. Check your answer graphically X^2

In this exercise we examine the effects of overbooking in the airline industry. Ontario Gateway Airlines'...

In this exercise we examine the effects of overbooking in the airline industry. Ontario Gateway Airlines' first class cabins have 10 seats in each plane. Ontario's overbooking policy is to sell up to 11 first class tickets, since cancellations and no-shows are always possible (and indeed are quite likely). For a given flight on Ontario Gateway, there were 11 first class tickets sold. Suppose that each of the 11 persons who purchased tickets has a 20% chance of not showing...

In this exercise, we examine the effect of combining investments with positively correlated risks, negatively correlated...

In this exercise, we examine the effect of combining investments with positively correlated risks, negatively correlated risks, and uncorrelated risks. A firm is considering a portfolio of assets. The portfolio is comprised of two assets, which we will call ''A" and "B." Let X denote the annual rate of return from asset A in the following year, and let Y denote the annual rate of return from asset B in the following year. Suppose that E(X) = 0.15 and E(Y)...

Question