DOCSLIB.ORG

Math 163 — R E V I E W E X a M 3 — Fall 2016 Topics Covered 1

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math 163 — R E V I E W E X a M 3 — Fall 2016 Topics Covered 1 MATH 163 | R E V I E W E X A M 3 | FALL 2016 TOPICS COVERED 1. Sequences fa g1 n n=n0 ◦ Find limits of sequences ◦ Graph sequences 1 P 2. Series an n=n0 ◦ Know the definition of the value of a series in terms of the limit of partial sums. Distinguish between the sequence of summands fa g1 , and the sequence of partial sums fs g1 , n n=n0 n n=n0 where s = Pn a . n k=n0 k ◦ Define divergence, convergence, and absolute convergence of a series. Know examples of series that diverge, converge absolutely, and converge conditionally. ◦ Be able to evaluate converging - geometric series (this is the Taylor series for f(x) = 1=(1 − x)) - telescoping series - series that happen to be other known Taylor series for functions such as ex; cos(x); or sin(x), at particular values of x. P1 n P1 n P1 n Examples: n=0 1=2 , n=0 (−10) =n!, n=0 n(x − 1) , ◦ Be able to determine whether a series diverges, converges absolutely, or converges condition- ally, using the Divergence Test, Limit Comparison Test, Direct Comparison Test, Ratio Test, Integral Test, Alternating Series Test. ◦ Be able to estimate series by partial sums using the alternating series estimation test. Do you know of any other methods to estimate series? 1 P n 3. Power Series cn(x − a) , power series representation of functions n=0 ◦ Find radius and interval of convergence of a power series. P1 n ◦ Uniqueness. The power series representation n=0 cn(x − a) of a function about the base point x = a is unique: if f has a power series representation about x = a that converges to f in (a − R; a + R), for some R > 0, then the unique coefficients are given by f (n)(a) c = n n! 4. Taylor series ◦ Know how to find Taylor series for any function either using the definition 1 X f (n)(c) f 00(c) f 000(3) (x − a)nf(c) + f 0(c)(x − a) + (x − a)2 + (x − a)3 + ::: n! 2 3! n=0 or using known Taylor series (e.g. for 1=(1 − x); ex; sin(x); cos(x) and substitution addition, multiplication differentiation integration ◦ When can you add, multiply, differentiate, integrate a series term-by-term? ◦ Memorize the Taylor series about x = 0 (the MacLaurin series) for 1 f(x) = ; f(x) = ex ; f(x) = sin(x) ; f(x) = cos(x) ; 1 − x and their intervals of convergence. ◦ Find Taylor polynomials pn(x) of degree n for given function f, about a base point x = a. ◦ Be able to approximate a function by a Taylor polynomial f(x) ≈ pn(x) and estimate the error Rn(x) made using - Taylor's Inequality. - the Alternating Series Estimation test. (n+1) f (c) n+1 Note: f(x) = pn(x) + Rn(x) where Rn(x) = (n+1)! (x − a) , where c is some (unknown) number between x and a. This holds, provided f is n + 1 times continuously differentiable between x and a. This formula is a generalization of the Mean Value Theorem, and we need it in order to determine how good the approximation of a function by a polynomial is. ◦ Note: the Taylor polynomial pn(x) for f(x) about x = a is the unique polynomial of degree n that satisfies that all its derivatives of order ≤ n agree with the derivatives of f at x = a, (k) (k) pn (a) = fn (a) ; k ≤ n For example, p1(x) is our already well-known linear approximation L(x) of f(x), which satisfies that L(a) = f(a) and L0(a) = f 0(a). PRACTICE PROBLEMS The topics listed above form a summary of the material you need to know. Read it and make sure you understand the main ideas. A set of sample problems is given below. 1. (a) What does it mean to state that limk!1 ak = L? P1 (b) What does it mean to state that k=1 ak = L? P1 Pn (c) Assume k=1 ak = L. What is limk!1 ak? What is limn!1 sn, where sn = k=1 ak? k (d) For ak = 1=2 , (i) sketch a plot of the sequence fakg Pn (ii) sketch a plot of the sequence fsng, where sn = k=1 ak? P1 (iii) Find k=1 ak. 3. What is a telescoping series? Give two examples of telescoping series and determine their value, or determine that they diverge. 4. Consider the geometric series P1 rk. Give two examples of geometric series, one starting k=k1 at k1 = 0, another starting at k1 6= 0, and find their values. 5. What is an absolutely converging series? What is a conditionally converging series? Give an example of each. 6. Fill in the blanks using either may or must. Explain your answers. (a) A series with summands tending to 0 converge. (b) A series that converges have summands that tend to zero. (c) If a series diverges, then the summands not tend to 0. (d) If a series diverges, then the Divergence Test succeed in proving the divergence. P1 P1 (e) If n=10 an diverges, then n=1000 an diverge. P1 P1 (f) If n=10 an converges, then n=10 janj converge. P1 P1 (g) If n=10 janj converges, then n=10 an converge. 7. (a) REVIEW Chapter 11 (pp 802), Concept Check: 9 1 X xn (b) Let f(x) = . Find series representations for f 0, f 00. Find the intervals of conver- n2 n=1 gence of f, f 0, and f 00. (Make sure you use your answer to (a)!) 8. REVIEW Chapter 11 (pp 802-804), True-False: 6,9,10,11,17,18,19. Explain your answers. P1 n 9. Suppose n=1 an6 diverges. What can you say about P1 n (a) the series n=1 an2 ? P1 (b) the series n=1 an? P1 n (c) the series n=1 an8 ? P1 n (d) the series n=1 an(−3) ? P1 n (e) the radius of convergence of the power series n=1 an(x − a) ? 10. Evaluate the following series or determine that they diverge: Review Chapter 11 (p 803), 27-31. 11. Determine whether the following series converge conditionally, converge absolutely, or diverge. Review Chapter 11 (p 803), 11,12,13,14,15,16. 12. Find the first 5 nonzero terms of the Taylor series of f(x) = sin x at a = π=6. State the linear approximation of f about a = π=6. State the Taylor polynomial of degree 2 p2 for f about a = π=6. How large is the magnitude of the error in the approximation f(x) ≈ p2(x), for x 2 [0; π=3], at most? 13. REVIEW Chapter 11 (p 804), # 47,48,50,51. (Find the Maclaurin series for f and its radius of convergence.) 14. Determine the radius and the interval of convergence of the following power series. 1 X 3n(x + 4)n p n n=1 15. REVIEW Chapter 11 (p 804), # 56 (approximate integral) Note: \correct to two decimals" means to within an error of 0.005. 16. (a) Write down a formula for the Taylor series of f(x) about a basepoint x = a. (b) Write down a formula for the Taylor polynomial pn(x) of f(x) about x = a. (c) If pn is used to approximate a function f about x = a, f(x) ≈ pn(x) Write down an upper bound for the error jf(x) − pn(x)j. x (d) Find p2(x) for f(x) = e about x = 0. Write down a formula for an upper bound for jf(x) − p2(x)j. Can you find an upper bound for jf(x) − p2(x)j if jxj < :1?.
Load more

Recommended publications
  • 3.3 Convergence Tests for Infinite Series
    3.3 Convergence Tests for Infinite Series 3.3.1 The integral test We may plot the sequence an in the Cartesian plane, with independent variable n and dependent variable a: n X The sum an can then be represented geometrically as the area of a collection of rectangles with n=1 height an and width 1. This geometric viewpoint suggests that we compare this sum to an integral. If an can be represented as a continuous function of n, for real numbers n, not just integers, and if the m X sequence an is decreasing, then an looks a bit like area under the curve a = a(n). n=1 In particular, m m+2 X Z m+1 X an > an dn > an n=1 n=1 n=2 For example, let us examine the first 10 terms of the harmonic series 10 X 1 1 1 1 1 1 1 1 1 1 = 1 + + + + + + + + + : n 2 3 4 5 6 7 8 9 10 1 1 1 If we draw the curve y = x (or a = n ) we see that 10 11 10 X 1 Z 11 dx X 1 X 1 1 > > = − 1 + : n x n n 11 1 1 2 1 (See Figure 1, copied from Wikipedia) Z 11 dx Now = ln(11) − ln(1) = ln(11) so 1 x 10 X 1 1 1 1 1 1 1 1 1 1 = 1 + + + + + + + + + > ln(11) n 2 3 4 5 6 7 8 9 10 1 and 1 1 1 1 1 1 1 1 1 1 1 + + + + + + + + + < ln(11) + (1 − ): 2 3 4 5 6 7 8 9 10 11 Z dx So we may bound our series, above and below, with some version of the integral : x If we allow the sum to turn into an infinite series, we turn the integral into an improper integral.
    [Show full text]
  • On a Series of Goldbach and Euler Llu´Is Bibiloni, Pelegr´I Viader, and Jaume Parad´Is
    On a Series of Goldbach and Euler Llu´ıs Bibiloni, Pelegr´ı Viader, and Jaume Parad´ıs 1. INTRODUCTION. Euler’s paper Variae observationes circa series infinitas [6] ought to be considered important for several reasons. It contains the first printed ver- sion of Euler’s product for the Riemann zeta-function; it definitely establishes the use of the symbol π to denote the perimeter of the circle of diameter one; and it introduces a legion of interesting infinite products and series. The first of these is Theorem 1, which Euler says was communicated to him and proved by Goldbach in a letter (now lost): 1 = 1. n − m,n≥2 m 1 (One must avoid repetitions in this sum.) We refer to this result as the “Goldbach-Euler Theorem.” Goldbach and Euler’s proof is a typical example of what some historians consider a misuse of divergent series, for it starts by assigning a “value” to the harmonic series 1/n and proceeds by manipulating it by substraction and replacement of other series until the desired result is reached. This unchecked use of divergent series to obtain valid results was a standard procedure in the late seventeenth and early eighteenth centuries. It has provoked quite a lot of criticism, correction, and, why not, praise of the audacity of the mathematicians of the time. They were led by Euler, the “Master of Us All,” as Laplace christened him. We present the original proof of the Goldbach- Euler theorem in section 2. Euler was obviously familiar with other instances of proofs that used divergent se- ries.
    [Show full text]
  • Series, Cont'd
    Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 5 Notes These notes correspond to Section 8.2 in the text. Series, cont'd In the previous lecture, we defined the concept of an infinite series, and what it means for a series to converge to a finite sum, or to diverge. We also worked with one particular type of series, a geometric series, for which it is particularly easy to determine whether it converges, and to compute its limit when it does exist. Now, we consider other types of series and investigate their behavior. Telescoping Series Consider the series 1 X 1 1 − : n n + 1 n=1 If we write out the first few terms, we obtain 1 X 1 1 1 1 1 1 1 1 1 − = 1 − + − + − + − + ··· n n + 1 2 2 3 3 4 4 5 n=1 1 1 1 1 1 1 = 1 + − + − + − + ··· 2 2 3 3 4 4 = 1: We see that nearly all of the fractions cancel one another, revealing the limit. This is an example of a telescoping series. It turns out that many series have this property, even though it is not immediately obvious. Example The series 1 X 1 n(n + 2) n=1 is also a telescoping series. To see this, we compute the partial fraction decomposition of each term. This decomposition has the form 1 A B = + : n(n + 2) n n + 2 1 To compute A and B, we multipy both sides by the common denominator n(n + 2) and obtain 1 = A(n + 2) + Bn: Substituting n = 0 yields A = 1=2, and substituting n = −2 yields B = −1=2.
    [Show full text]
  • 3.2 Introduction to Infinite Series
    3.2 Introduction to Infinite Series Many of our infinite sequences, for the remainder of the course, will be defined by sums. For example, the sequence m X 1 S := : (1) m 2n n=1 is defined by a sum. Its terms (partial sums) are 1 ; 2 1 1 3 + = ; 2 4 4 1 1 1 7 + + = ; 2 4 8 8 1 1 1 1 15 + + + = ; 2 4 8 16 16 ::: These infinite sequences defined by sums are called infinite series. Review of sigma notation The Greek letter Σ used in this notation indicates that we are adding (\summing") elements of a certain pattern. (We used this notation back in Calculus 1, when we first looked at integrals.) Here our sums may be “infinite”; when this occurs, we are really looking at a limit. Resources An introduction to sequences a standard part of single variable calculus. It is covered in every calculus textbook. For example, one might look at * section 11.3 (Integral test), 11.4, (Comparison tests) , 11.5 (Ratio & Root tests), 11.6 (Alternating, abs. conv & cond. conv) in Calculus, Early Transcendentals (11th ed., 2006) by Thomas, Weir, Hass, Giordano (Pearson) * section 11.3 (Integral test), 11.4, (comparison tests), 11.5 (alternating series), 11.6, (Absolute conv, ratio and root), 11.7 (summary) in Calculus, Early Transcendentals (6th ed., 2008) by Stewart (Cengage) * sections 8.3 (Integral), 8.4 (Comparison), 8.5 (alternating), 8.6, Absolute conv, ratio and root, in Calculus, Early Transcendentals (1st ed., 2011) by Tan (Cengage) Integral tests, comparison tests, ratio & root tests.
    [Show full text]
  • Calculus Online Textbook Chapter 10
    Contents CHAPTER 9 Polar Coordinates and Complex Numbers 9.1 Polar Coordinates 348 9.2 Polar Equations and Graphs 351 9.3 Slope, Length, and Area for Polar Curves 356 9.4 Complex Numbers 360 CHAPTER 10 Infinite Series 10.1 The Geometric Series 10.2 Convergence Tests: Positive Series 10.3 Convergence Tests: All Series 10.4 The Taylor Series for ex, sin x, and cos x 10.5 Power Series CHAPTER 11 Vectors and Matrices 11.1 Vectors and Dot Products 11.2 Planes and Projections 11.3 Cross Products and Determinants 11.4 Matrices and Linear Equations 11.5 Linear Algebra in Three Dimensions CHAPTER 12 Motion along a Curve 12.1 The Position Vector 446 12.2 Plane Motion: Projectiles and Cycloids 453 12.3 Tangent Vector and Normal Vector 459 12.4 Polar Coordinates and Planetary Motion 464 CHAPTER 13 Partial Derivatives 13.1 Surfaces and Level Curves 472 13.2 Partial Derivatives 475 13.3 Tangent Planes and Linear Approximations 480 13.4 Directional Derivatives and Gradients 490 13.5 The Chain Rule 497 13.6 Maxima, Minima, and Saddle Points 504 13.7 Constraints and Lagrange Multipliers 514 CHAPTER Infinite Series Infinite series can be a pleasure (sometimes). They throw a beautiful light on sin x and cos x. They give famous numbers like n and e. Usually they produce totally unknown functions-which might be good. But on the painful side is the fact that an infinite series has infinitely many terms. It is not easy to know the sum of those terms.
    [Show full text]
  • INFINITE SERIES SERIES and PARTIAL SUMS What If We Wanted
    INFINITE SERIES SERIES AND PARTIAL SUMS What if we wanted to sum up the terms of this sequence, how many terms would I have to use? 1, 2, 3, . 10, . ? Well, we could start creating sums of a finite number of terms, called partial sums, and determine if the sequence of partial sums converge to a number. What do you think that this sequence of partial sums is converging to? It is approaching the value of 2. Therefore, we can conclude that the sum of all the terms of this sequence is 2. To discuss this topic fully, let us define some terms used in this and the following sets of supplemental notes. DEFINITION: Given a sequence of numbers {a n }, the sum of the terms of this sequence, a 1 + a 2 + a 3 + . + a n + . , is called an infinite series. DEFINITION: FACT: If the sequence of partial sums converge to a limit L, then we can say that the series converges and its sum is L. FACT: If the sequence of partial sums of the series does not converge, then the series diverges. There are many different types of series, but we going to start with series that we might of seen in Algebra. GEOMETRIC SERIES DEFINITION: FACT: FACT: If | r | 1, then the geometric series will diverge. EXAMPLE 1: Find the nth partial sum and determine if the series converges or diverges. SOLUTION: Now to calculate the sum for this series. EXAMPLE 2: Find the nth partial sum and determine if the series converges or diverges. 1 - 3 + 9 - 27 + .
    [Show full text]
  • A SIMPLE COMPUTATION of Ζ(2K) 1. Introduction in the Mathematical Literature, One Finds Many Ways of Obtaining the Formula
    A SIMPLE COMPUTATION OF ζ(2k) OSCAR´ CIAURRI, LUIS M. NAVAS, FRANCISCO J. RUIZ, AND JUAN L. VARONA Abstract. We present a new simple proof of Euler's formulas for ζ(2k), where k = 1; 2; 3;::: . The computation is done using only the defining properties of the Bernoulli polynomials and summing a telescoping series, and the same method also yields integral formulas for ζ(2k + 1). 1. Introduction In the mathematical literature, one finds many ways of obtaining the formula 1 X 1 (−1)k−122k−1π2k ζ(2k) := = B ; k = 1; 2; 3;:::; (1) n2k (2k)! 2k n=1 where Bk is the kth Bernoulli number, a result first published by Euler in 1740. For example, the recent paper [2] contains quite a complete list of references; among them, the articles [3, 11, 12, 14] published in this Monthly. The aim of this paper is to give a new proof of (1) which is simple and elementary, in the sense that it involves only basic one variable Calculus, the Bernoulli polynomials, and a telescoping series. As a bonus, it also yields integral formulas for ζ(2k + 1) and the harmonic numbers. 1.1. The Bernoulli polynomials | necessary facts. For completeness, we be- gin by recalling the definition of the Bernoulli polynomials Bk(t) and their basic properties. There are of course multiple approaches one can take (see [7], which shows seven ways of defining these polynomials). A frequent starting point is the generating function 1 xext X xk = B (t) ; ex − 1 k k! k=0 from which, by the uniqueness of power series expansions, one can quickly obtain many of their basic properties.
    [Show full text]
  • In Praise of an Elementary Identity of Euler Arxiv:1102.0659V3 [Math.CO] 12 Jun 2011
    In Praise of an Elementary Identity of Euler Gaurav Bhatnagar Educomp Solutions Ltd. [email protected] June 7, 2011 Dedicated to S. B. Ekhad and D. Zeilberger Abstract We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated ver- sion of this identity that we call Euler's Telescoping Lemma, we give alternate proofs of all the key summation theorems for terminating Hypergeometric Series and Basic Hypergeometric Series, including the terminating Binomial Theorem, the Chu{Vandermonde sum, the Pfaff–Saalsch¨utzsum, and their q-analogues. We also give a proof of Jackson's q-analog of Dougall's sum, the sum of a terminating, bal- anced, very-well-poised 8φ7 sum. Our proofs are conceptually the same as those obtained by the WZ method, but done without using a computer. We survey iden- tities for Generalized Hypergeometric Series given by Macdonald, and prove several identities for q-analogs of Fibonacci numbers and polynomials and Pell numbers that have appeared in combinatorial contexts. Some of these identities appear to be new. arXiv:1102.0659v3 [math.CO] 12 Jun 2011 Keywords: Telescoping, Fibonacci Numbers, Pell Numbers, Derangements, Hy- pergeometric Series, Fibonacci Polynomials, q-Fibonacci Numbers, q-Pell numbers, Basic Hypergeometric Series, q-series, Binomial Theorem, q-Binomial Theorem, Chu{Vandermonde sum, q-Chu{Vandermonde sum, Pfaff–Saalsch¨utzsum, q-Pfaff– Saalsch¨utzsum, q-Dougall summation, very-well-poised 6φ5 sum, Generalized Hy- pergeometric Series, WZ Method. MSC2010: Primary 33D15; Secondary 11B39, 33C20, 33F10, 33D65 1 1 Introduction One of the first results in q-series is Euler's 1740 expansion of the product (1 − q)(1 − q2)(1 − q3)(1 − q4) ··· into a power series in q.
    [Show full text]
  • MATH 341: TAKEAWAYS We Used a Variety of Results and Techniques from 103 and 104: (1) Standard Integration Theory: for Us, the M
    MATH 341: TAKEAWAYS STEVEN J. MILLER ABSTRACT. Below we summarize some items to take away from the class (as well as previous classes!). In particular, what are one time tricks and methods, and what are general techniques to solve a variety of problems, as well as what have we used from various classes. Comments and additions welcome! 1. CALCULUS I AND II (MATH 103 AND 104) We used a variety of results and techniques from 103 and 104: (1) Standard integration theory: For us, the most important technique is integration by parts; one of many places we used this was in computing the moments of the Gaussian. Integration by parts is a very powerful technique, and is frequently used. While most of the time it is clear how to choose the functions u and dv, sometimes we need to beR a bit clever. For exam- ¡1=2 1 2 2 ple, consider the second moment of the standard normal: (2¼) ¡1 x exp(¡x =2)dx. The natural choices are to take u = x2 or u = exp(¡x2=2), but neither of these work as they lead to choices for dv that do not have a closed form integral. What we need to do is split the two ‘natural’ functions up, and let u = x and dv = exp(¡x2=2)xdx. The reason is that while there is no closed form expression for the anti-derivative of the standard normal, once we have xdx instead of dx then we can obtain nice integrals. One final remark on integrating by parts: it is a key ingredient in the ‘Bring it over’ method (which will be discussed below).
    [Show full text]
  • 6.1 Sigma Notation & Convergence / Divergence
    Math 123 - Shields Infinite Series Week 6 Zeno of Elea posed some philosophical problems in the 400s BC. His actual reasons for doing so are somewhat murky, but they are troubling regardless. In his Dichotomy paradox he argues that any objects in motion must arrive at the halfway point before they arrive at their goal. However, we can iterate the reasoning, and an object must also arrive at the halfway point to the halfway point first, and so on :::. If the goal is m meters away and you travel at a speed of v, then the time needed to traverse the distance is x x x x t = + + + + ::: 2v 4v 8v 16v x1 1 1 1 = + + + + ::: v 2 4 8 16 The parenthetical expression is an infinite sum. Zeno said that the time for the motion then must be infinite since if we add infinitely many positive numbers together then this must result in a positive number. As beings of logic we are forced to conclude that motion is an illusion. We have all been duped. On a completely unrelated note recall from Calc I that for small angles θ we can use a local linear approxi- mation to eθ. That is eθ ≈ 1 + θ This approximation is pretty good, but it's not perfect because the approximation is a line and eθ bends. Well in a similar way we can make our estimate better by taking a quadratic approximation and say 1 eθ ≈ 1 + θ + θ2 2 This is better, but still ultimately an approximation with some built in error.
    [Show full text]
  • On Series Involving Fibonacci and Lucas Numbers I 1 Introduction
    On series involving Fibonacci and Lucas numbers I Daniel Duverney and Iekata Shiokawa 1 Introduction This paper is the first of two survey papers devoted to the study of series involving Fibonacci and Lucas numbers. There has been, up to now, a great deal of works on this subject, and it seems necessary to summarize and classify it. To begin with, we observe that series involving Fibonacci and Lucas numbers can be divided, roughly, into two large classes ; first, the subscript of Fibonacci and Lucas numbers appear in arithmetic progression ; second, they appear in geometrie progression, We give here two classical examples of both cases. For these two exmples, the sum of the series can be explicitly computed. Example 1.1. We have (1.1) 1+V5 . where <I>= 2 lS the Golden Number. It seems difficult to find the first mention of this result, which is a direct conse- quence of the well-known continued fraction expansion of 'If (see for example [15], Exercise 3.11). But it is undoubtedly very old. Example 1.2. Lucas found in 1878 [27] that ~_1 = 7-V5. (1.2) ~F2n 2 n=O This result can be easily obtained by taking x = 'If in de Morgan's series (cf. [9]) x ---, (1.3) 1-x which is readily deduced from the identity 2 x X x --- (1.4) 1- x 1- x2· 1 There are three natural questions about senes involving Fibonacci and Lucas numbers. First question: Given a series, can we compute its sum? That means, can we express this series in a closed form as in Examples 1.1 and 1'.2, or alternatively, can we express it by using classical functions? These will include, as will see later, q-hypergeometric functions, Lambert series, elliptic integrals, and theta functions.
    [Show full text]
  • Review Solutions: Chapter 11
    Review Solutions: Chapter 11 1. What does it mean to say that a series converges? th SOLUTION: We define the n partial sum sn as follows: n X S1 = a1 S2 = a1 + a2 S3 = a1 + a2 + a3 ··· Sn = an k=1 The partial sums Sn form a sequence (of numbers). The (infinite) series is said to converge to a sum S if and only if the limit of the partial sums is S. That is, if the limit lim Sn = S n!1 then the series is said to converge, and the sum is said to be equal to S: 1 X an = S n=1 2. Does the given sequence or series converge or diverge? 1 1 (a) X p n=2 n − n P 1 SOLUTION: Using the dominating terms, this looks a lot like n , so we use the limit comparison (note that both series, the one given and the template, have all positive terms) 1p n 1 lim n− n = lim p = lim = 1 n!1 1 n!1 n!1 p1 n n − n 1 − n Therefore, both series diverge together (by the limit comparison test). (b) Take the limit: n + 2 n + 2 lim ln = ln lim = ln(1) = 0 n!1 n n!1 n n o np (c) 1+ n SOLUTION: Take the limit; You can use L'Hospital's rule if you like. To be precise, we ought to change notation to x (since you cannot formally take the derivative of a discrete sequence): x 1 p lim p = lim p = lim 2 x = 1 x!1 1 + x x!1 1=2 x x!1 Therefore, the sequence diverges.
    [Show full text]