Divergence Test

10 Key excerpts on "Divergence Test"

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  A Radical Approach to Real Analysis

  Second Edition

  • David Bressoud(Author)
  • 2006(Publication Date)
  • American Mathematical Society

    (Publisher)

  Its justification rests on the Cauchy criterion. 130 4 The Convergence of Infinite Series Theorem 4.6 (The Comparison Test). Let a 1 + a 2 + a 3 + · · · and b 1 + b 2 + b 3 + · · · be two series with summands that are greater than or equal to zero. We assume that each b i is greater than or equal to the corresponding a i : b 1 ≥ a 1 ≥ 0 , b 2 ≥ a 2 ≥ 0 , b 3 ≥ a 3 ≥ 0 , . . . If b 1 + b 2 + b 3 + · · · converges, then so does a 1 + a 2 + a 3 + · · · . If a 1 + a 2 + a 3 + · · · diverges, then so does b 1 + b 2 + b 3 + · · · . Proof: Let S n = a 1 + a 2 + · · · + a n and T n = b 1 + b 2 + · · · + b n . If m < n , then 0 ≤ S n − S m = a m + 1 + a m + 2 + · · · + a n ≤ b m + 1 + b m + 2 + · · · + b n = T n − T m , and so | S n − S m | ≤ | T n − T m | . (4.8) We assume the series b 1 + b 2 + b 3 + · · · converges. Given a positive bound , we have a response N . Equation (4.8) shows us that the same response will work for the series a 1 + a 2 + a 3 + · · · . The contrapositive of what we have just proven says that if a 1 + a 2 + a 3 + · · · diverges, then b 1 + b 2 + b 3 + · · · diverges. Q.E.D. The Ratio Test The ratio and root test rely on comparing our series to a geometric series. They are very simple and powerful techniques that quickly yield one of three conclusions: 1. the series in question converges absolutely, 2. the series in question diverges, or 3. the results of this test are inconclusive. It is the third possibility that is the principal drawback of these tests. The most interesting series mathematicians and scientists were encountering in the early 1800s all fell into category 3. Nevertheless, these tests are important because they are simple. Start with one of these tests, and move on to a more complicated test only if the results are inconclusive. Theorem 4.7 (The Ratio Test). Given a series with nonzero summands, a 1 + a 2 + a 3 + · · · , we consider the ratio r ( n ) = | a n + 1 /a n | .

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  Mathematical Methods for Physicists

  • George B. Arfken(Author)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  5.1 Comparison tests EXAMPLE 5.21 The p Series Test Ση η ~ Ρ > P = 0.999, for convergence. Since n ~ 0 9 > n~ and b n = n _1 forms the divergent harmonic series, the comparison test shows that X„«~ 0 ' 999 is divergent. Generalizing, £„ n~ p is seen to be divergent for all p < 1. Cauchy Root Test If (a n ) 1,n < r < 1 for all sufficiently large n, with r independent of n, then £ n a n is convergent. If (a n ) lln > 1 for all sufficiently large n, then £„ a n is divergent. 282 INFINITE SERIES The first part of this test is verified easily by raising ( „) 1/ < r to the nth power. We get a n < r n < 1. Since r n is just the nth term in a convergent geometric series, ^„a„ is convergent by the comparison test. Conversely, if (a n ) lln > 1, then a n > 1 and the series must diverge. This root test is particularly useful in establishing the properties of power series (Section 5.7). D'Alembert or Cauchy Ratio Test If a n+1 /a n < r < 1 for all sufficiently large n, and r is independent of n, then Σ„α η is convergent. If a n+1 /a n > 1 for all sufficiently large n, then Σ„α η is divergent. Convergence is proved by direct comparison with the geometric series (1 + r + r 2 + · · ·). In the second part a n+1 >a n and divergence should be reasonably obvious. Although not quite so sensitive as the Cauchy root test, this D'Alembert ratio test is one of the easiest to apply and is widely used. An alternate statement of the ratio test is in the form of a limit: If lim -^ < 1, convergence, > 1, divergence, (5.16) = 1, indeterminant. Because of this final indeterminant possibility, the ratio test is likely to fail at crucial points, and more delicate, more sensitive tests are necessary. The alert reader may wonder how this indeterminacy arose. Actually it was concealed in the first statement a n+l /a n < r < 1. We might encounter a n+1 /a n < 1 for all finite n but be unable to choose an r < 1 and independent ofn such that a n+l /a n < r for all sufficiently large n.

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  Calculus

  Single Variable

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  Later, we will use some of these tests to study the convergence of Taylor series. The Comparison Test We will begin with a test that is useful in its own right and is also the building block for other important convergence tests. The underlying idea of this test is to use the known convergence or divergence of a series to deduce the convergence or divergence of another series. 9.5 The Comparison, Ratio, and Root Tests 535 Theorem 9.5.1: Comparison Test Let ∑ ∞ k=1 a k and ∑ ∞ k=1 b k be series with nonnegative terms and suppose that a 1 ≤ b 1 , a 2 ≤ b 2 , a 3 ≤ b 3 , . . . , a k ≤ b k , . . . (a) If the “bigger series” Σb k converges, then the “smaller series” Σa k also converges. (b) If the “smaller series” Σa k diverges, then the “bigger series” Σb k also diverges. In Theorem 9.5.1, it is not essential that the condition a k ≤ b k hold for all k, as stated; the conclusions of the theorem remain true if this condition is eventually true. We have left the proof of this theorem for the exercises; however, it is easy to visualize why the theorem is true by interpreting the terms in the series as areas of rectangles (Figure 9.5.1). The comparison test states that if the total area ∑ b k is finite, then the total area ∑ a k must also be finite; and if the total area ∑ a k is infinite, then the total area ∑ b k must also be infinite. 1 . . . . . . . . . . . . a 1 b 1 2 a 2 b 2 3 a 3 b 3 4 a 4 b 4 5 a 5 b 5 k a k b k For each rectangle, a k denotes the area of the blue portion and b k denotes the combined area of the white and blue portions. FIGURE 9.5.1 Using the Comparison Test There are two steps required for using the comparison test to determine whether a series ∑ u k with positive terms converges: Step 1. Guess whether the series ∑ u k converges or diverges. Step 2. Find a series that proves the guess to be correct.

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  Calculus Early Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  Later, we will use some of these tests to study the convergence of Taylor series. The Comparison Test We will begin with a test that is useful in its own right and is also the building block for other important convergence tests. The underlying idea of this test is to use the known convergence or divergence of a series to deduce the convergence or divergence of another series. In Theorem 10.5.1, it is not essential that the condition a k ≤ b k hold for all k, as stated; the conclusions of the theorem remain true if this condition is eventually true. Theorem 10.5.1: Comparison Test Let ∑ ∞ k=1 a k and ∑ ∞ k=1 b k be series with nonneg- ative terms and suppose that a 1 ≤ b 1 , a 2 ≤ b 2 , a 3 ≤ b 3 , …, a k ≤ b k , … (a) If the “bigger series” ∑ b k converges, then the “smaller series” ∑ a k also con- verges. (b) If the “smaller series” ∑ a k diverges, then the “bigger series” ∑ b k also diverges. We have left the proof of this theorem for the exercises; however, it is easy to visualize why the theorem is true by interpreting the terms in the series as areas of rectangles (Figure 10.5.1). The comparison test states that if the total area ∑ b k is finite, then the total area ∑ a k must also be finite; and if the total area ∑ a k is infinite, then the total area ∑ b k must also be infinite. 1 . . . . . . . . . . . . a 1 b 1 2 a 2 b 2 3 a 3 b 3 4 a 4 b 4 5 a 5 b 5 k a k b k For each rectangle, a k denotes the area of the blue portion and b k denotes the combined area of the white and blue portions. ▴ Figure 10.5.1 Using the Comparison Test There are two steps required for using the comparison test to determine whether a series ∑ u k with positive terms converges: Step 1. Guess whether the series ∑ u k converges or diverges. Step 2. Find a series that proves the guess to be correct. That is, if we guess that ∑ u k diverges, we must find a divergent series whose terms are “smaller” than

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  Calculus

  Early Transcendental Single Variable

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  The underlying idea of this test is to use the known convergence or divergence of a series to deduce the convergence or divergence of another series. It is not essential in Theorem 9.5.1 that the condition a k ≤ b k hold for all k, as stated; the conclusions of the theorem remain true if this condition is eventu- ally true. 9.5.1 THEOREM (The Comparison Test) Let ∑ ∞ k=1 a k and ∑ ∞ k=1 b k be series with non- negative terms and suppose that a 1 ≤ b 1 , a 2 ≤ b 2 , a 3 ≤ b 3 , . . . , a k ≤ b k , . . . (a) If the “bigger series” Σb k converges, then the “smaller series” Σa k also con- verges. (b) If the “smaller series” Σa k diverges, then the “bigger series” Σb k also diverges. We have left the proof of this theorem for the exercises; however, it is easy to visual- ize why the theorem is true by interpreting the terms in the series as areas of rectangles (Figure 9.5.1). The comparison test states that if the total area ∑ b k is finite, then the total Figure 9.5.1 area ∑ a k must also be finite; and if the total area ∑ a k is infinite, then the total area ∑ b k must also be infinite. USING THE COMPARISON TEST There are two steps required for using the comparison test to determine whether a series ∑ u k with positive terms converges: Step 1. Guess at whether the series ∑ u k converges or diverges. Step 2. Find a series that proves the guess to be correct. That is, if we guess that ∑ u k diverges, we must find a divergent series whose terms are “smaller” than the corresponding terms of ∑ u k , and if we guess that ∑ u k converges, we must find a convergent series whose terms are “bigger” than the corresponding terms of ∑ u k . In most cases, the series ∑ u k being considered will have its general term u k expressed as a fraction. To help with the guessing process in the first step, we have formulated two principles that are based on the form of the denominator for u k . These principles sometimes suggest whether a series is likely to converge or diverge.

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  Calculus

  Concepts and Contexts, Enhanced Edition

  • James Stewart(Author)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 580 CHAPTER 8 INFINITE SEQUENCES AND SERIES Note: The terms of the series being tested must be smaller than those of a convergent series or larger than those of a divergent series. If the terms are larger than the terms of a convergent series or smaller than those of a divergent series, then the Comparison Test doesn’t apply. Consider, for instance, the series The inequality is useless as far as the Comparison Test is concerned because is convergent and . Nonetheless, we have the feeling that ought to be convergent because it is very similar to the convergent geometric series . In such cases the fol-lowing test can be used. The Limit Comparison Test Suppose that and are series with positive terms. If where c is a finite number and , then either both series converge or both diverge. Although we won’t prove the Limit Comparison Test, it seems reasonable because for large . Using the Limit Comparison Test Test the series for convergence or divergence. SOLUTION We use the Limit Comparison Test with and obtain Since this limit exists and is a convergent geometric series, the given series con-verges by the Limit Comparison Test. Estimating the Sum of a Series Suppose we have been able to use the Integral Test to show that a series is convergent and we now want to find an approximation to the sum of the series. Of course, any par-tial sum is an approximation to because .

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  Infinite Series

  • Isidore Isaac Hirschman(Author)
  • 2014(Publication Date)
  • Dover Publications

    (Publisher)

  [ 1 ]

  Tests for Convergence and Divergence

  1. Sequences

       Since infinite series are studied by means of sequences, this first section is devoted to a very brief review of sequences. An infinite sequence is simply a succession of numbers,

  indexed by consecutive integers. Here the “...” always written in just this way stands for the remaining terms: s4 , s5 , etc., sn being the term of index n. Alternatively we can write

  A sequence need not begin with the term of index 1 but may start with any index. In more technical language a sequence whose initial index is m,

  is a real valued function on the set (m, m + 1, m + 2, ···), the value of this function at n being tn .

  We shall be particularly concerned with limits. In the first part of this section our discussion will be somewhat informal. A parallel formal development of this same subject is given Sec. 1* which follows.

  DEFINITION 1a. Let sn (n = 1, 2, ···) be a sequence. We say that the limit of the sequence sn (n = 1, 2, ···) is s, equivalently , if sn is arbitrarily close to s for all sufficiently large n.

  For example, Some sequences do not have limits; for example, the sequences and

  do not have limits. When limits exist, they may be combined in various ways. The following discussion includes only a few of these ways, but these few are so important that they deserve special attention. Let

  be a sequence and c a constant. Then from these ingredients we can manufacture a new sequence whose terms are

  THEOREM 1b. If sn exists, then csn exists, and

  To see this, imagine that sn = 7 and c = 2. Then for n very large, sn is very near 7, and therefore 2sn is very near 14.

  Again given two sequences (with the same initial index) we can form a new sequence whose terms are

  THEOREM 1c. If sn and tn exist, so does (sn + tn ) and .

  Again, if one thinks of a special case, this result becomes obvious. For example suppose that sn = 7, tn = 11. Then if n is very large, sn is close to 7 and tu is close to 11, and thus sn + tn

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  Real Analysis

  With Proof Strategies

  • Daniel W. Cunningham(Author)
  • 2021(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  7

  Infinite Series

  Around 1665, Newton discovered that any binomial of the form

  ( 1 + x )

  r

  , where

  r ∉ ℕ

  and

  | x |

  < 1

  , can be expressed as an infinite series. In 1673, Leibniz discovered that π can be written as an infinite series. For Newton and Leibniz, infinite series were of fundamental importance in their development of calculus; however, during these times, questions of rigor and convergence were of secondary importance.

  One typically first studies infinite series, in a standard calculus course, without going deeply into the theory of infinite series. In this chapter, using the results in Chapter 3 on the convergence of sequences, we will prove the important theorems that concern an infinite series of real numbers. In Section 7.1, we introduce the concept of a convergent infinite series, and discuss a variety of different kinds of infinite series. We then examine in Section 7.2 a range of tests for convergence. Finally, in Section 7.3 we investigate the effect of regrouping and rearranging the terms of an infinite series.

  7.1 CONVERGENCE AND DIVERGENCE

  As we have seen, the completeness axiom and the limit operation are critical components in real analysis. In particular, these concepts are used to develop the derivative and the Riemann integral. The limit concept was also used in our study of sequences, and it will again be used in our examination of infinite series.

  Definition. (Infinite Series). Given a sequence

  〈

  a n

  〉

  of real numbers, the nth partial sums

  n

  is defined by

  s n = ∑ k = 1 n a k = a 1 + a 2 + a 3 + ⋯ + a n .

  We refer to the sequence

  〈

  s n

  〉

  of partial sums as the infinite series

  ∑

  k = 1

  ∞

  a k

  . The real numbers

  a 1

  ,

  a 2

  ,

  a 3

  , …

  are called the terms

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  Real and Complex Analysis

  • Christopher Apelian, Steve Surace(Authors)
  • 2009(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  To prove part b ), note that there exists a positive integer N such that for j > N , | x j +1 | | x j | -r < r -1 2 . This in turn implies that for j > N , | x j +1 | > 1 + r 2 | x j | , or equivalently, for t ≡ 1+ r 2 > 1, | x j | > t j -N -1 | x N +1 | for j ≥ N + 2. From this it follows that lim x j = 0. Therefore, by the test for divergence, the series ∑ ∞ j =1 x j diverges. Example 4.30 Fix z 0 ∈ C . We will show that the series ∑ ∞ n =0 z n 0 n ! converges absolutely by use of the ratio test. Simply consider the limit lim z n +1 0 / ( n + 1)! z n 0 /n ! = lim z 0 n + 1 = 0. Note that since z 0 ∈ C was arbitrary, this result holds for any z 0 ∈ C . The idea behind the truth of parts a ) and b ) of the ratio test is that, in each of these two cases, the tails of the given series can be seen to “look like” that of a convergent or a divergent geometric series, respectively. Since the convergence behavior of a series depends only on the behavior of its tail, we 122 LIMITS AND CONVERGENCE obtain a useful test for convergence in these two cases. Case c ) is a case that the ratio test is not fine enough to resolve. Other methods need to be tried in this case to properly determine the series” behavior. Similar reasoning gives rise to another useful test that is actually more general than the ratio test, although not always as easy to apply. That test is the root test. Our version of the root test defines ρ as a lim sup rather than as a limit, which gives the test more general applicability since the lim sup always exists (if we include the possible value of ∞ ), whereas the limit might not. Of course, when the limit does exist, it equals the lim sup . Theorem 4.31 (The Root Test) Let { x j } be a sequence of nonzero elements of X , and suppose lim sup j | x j | = ρ exists. Then, a ) ρ < 1 ⇒ ∑ ∞ j =1 | x j | converges, and hence, ∑ ∞ j =1 x j converges. b ) ρ > 1 ⇒ ∑ ∞ j =1 x j diverges. c ) ρ = 1 ⇒ The test is inconclusive.

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  A Mathematical Bridge

  An Intuitive Journey in Higher Mathematics

  • Stephen Hewson(Author)
  • 2009(Publication Date)
  • WSPC

    (Publisher)

  The question of the behaviour of the series on the radius of convergence is usually a very difficult one to answer, and varies wildly from series to series; anything can happen (Fig. 3.7). Convergence everywhere inside circle Convergence must be checked for each individual point D ivergence outside circle Fig. 3.7 The circle of convergence for a complex series. 148 A Mathematical Bridge 3.2.3.1 Determining the radius of convergence We can use the ratio test to try to determine the value of the radius of convergence for a power series S = ∑ ∞ n =0 a n z n : • lim n →∞ vextendsingle vextendsingle vextendsingle a n +1 z n +1 a n z n vextendsingle vextendsingle vextendsingle = lim n →∞ vextendsingle vextendsingle vextendsingle a n +1 a n vextendsingle vextendsingle vextendsingle | z | braceleftBigg < 1 ⇒ convergence = 1 ⇒ ? > 1 ⇒ divergence Although it is an obvious point, it is noteworthy that the fixed value of the limit of | a n +1 /a n | is independent of the choice of z . Suppose that this has some well defined limit. We can then use this to determine the radius of convergence • 1 R = lim n →∞ vextendsingle vextendsingle vextendsingle a n +1 a n vextendsingle vextendsingle vextendsingle (if the limit exists and is positive) As a special case, we define R = ∞ if the limit of the ratio is zero. This means that the power series converges for any value of z . At the other end of the spectrum, if the ratio of the limit is infinite then the radius of convergence of the series is zero: the series will diverge for any non-zero choice of z . 3.2.4 Rearrangement of infinite series We have really begun to make progress into the theory of limiting processes, and now have clear guidelines concerning the questions of convergence or divergence of a given infinite sequence.

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