Mathematics
P Series
In mathematics, a P series is a type of infinite series that takes the form Σ(1/n^p), where n ranges from 1 to infinity and p is a constant. The convergence of P series depends on the value of p; if p is greater than 1, the series converges, while if p is less than or equal to 1, the series diverges.
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4 Key excerpts on "P Series"
eBook - PDF- Ron Larson, Bruce Edwards(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
This is shown in the proof of Theorem 9.11. THEOREM 9.11 Convergence of p -Series The p-series ∑ ∞ n =1 1 n p = 1 1 p + 1 2 p + 1 3 p + 1 4 p + . . . converges for p > 1 and diverges for 0 < p ≤ 1. Proof The proof follows from the Integral Test and from Theorem 8.7, which states that integral.alt1 ∞ 1 1 x p dx converges for p > 1 and diverges for 0 < p ≤ 1. Convergent and Divergent p-Series Discuss the convergence or divergence of (a) the harmonic series and (b) the p-series with p = 2. Solution a. From Theorem 9.11, it follows that the harmonic series ∑ ∞ n =1 1 n = 1 1 + 1 2 + 1 3 + . . . p = 1 diverges. b. From Theorem 9.11, it follows that the p-series ∑ ∞ n =1 1 n 2 = 1 1 2 + 1 2 2 + 1 3 2 + . . . p = 2 converges. HARMONIC SERIES Pythagoras and his students paid close attention to the development of music as an abstract science. This led to the discovery of the relationship between the tone and the length of a vibrating string. It was observed that the most beautiful musical harmonies corresponded to the simplest ratios of whole numbers. Later mathematicians developed this idea into the harmonic series, where the terms in the harmonic series correspond to the nodes on a vibrating string that produce multiples of the fundamental frequency. For example, 1 2 is twice the fundamental frequency, 1 3 is three times the fundamental frequency, and so on. 612 Chapter 9 Infinite Series The sum of the series in Example 3(b) can be shown to be π 2 H208626. (This was proved by Leonhard Euler, but the proof is too difficult to present here.) Be sure you see that the Integral Test does not tell you that the sum of the series is equal to the value of the integral. For instance, the sum of the series in Example 3(b) is ∑ ∞ n =1 1 n 2 = π 2 6 ≈ 1.645 whereas the value of the corresponding improper integral is integral.alt1 ∞ 1 1 x 2 dx = 1. Testing a Series for Convergence Determine whether the series ∑ ∞ n =2 1 n ln n converges or diverges.
- Bernhard W. Bach, Jr.(Authors)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
Recall (from Section 2.3.1 ) that the harmonic series is a special case of the p -series ( p =1). The divergence test established that a p -series diverges for p < 1 but produced an indeterminate result for p = 1. With the use of the integral test, we now know that the p -series diverges for p ≤ 1. In the following section , we will make use of the comparison test to prove that the p -series is convergent for p > 1, thereby proving the p -series theorems, which were given without proof in Section 2.3.1 . 2.3.4 The Comparison Test The comparison test establishes the convergence or divergence of an unfami-liar positive term series by comparing it to a known (convergent or divergent) positive term series. For example, let the series 2.3 Testing In fi nite Series for Convergence 35 b 1 þ b 2 þ b 3 þ … ¼ X ∞ n ¼ 1 b n be a convergent positive term series, and let the series a 1 þ a 2 þ a 3 þ … ¼ X ∞ n ¼ 1 a n d be an unfamiliar series. If a n ≤ b n for all integers n beyond some point, then the series ∑ a n is convergent. Stated another way, a positive term series is con-vergent if it is dominated by a convergent series. The concept of dominance and how it relates to convergence are illustrated in Figure 2.2 . On the other hand, if the positive term series b 1 þ b 2 þ b 3 þ … ¼ X ∞ n ¼ 1 b n is known to be divergent and the terms of the unfamiliar series a 1 þ a 2 þ a 3 þ … ¼ X ∞ n ¼ 1 a n a n b n n Value of the n th term Figure 2.2 If ∑ b n is known to be convergent, then ∑ a n is convergent. 36 In fi nite Series are such that a n ≥ b n for all integers n beyond some point, then the series ∑ a n is divergent. Put another way, a positive term series is divergent if it dominates a divergent series. Figure 2.3 illustrates the concept of dominance and how it relates to divergence. Be careful to not misapply the comparison test. If an unfamiliar series dominates a convergent series, the unfamiliar series may be convergent or divergent.
- James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
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. 724 CHAPTER 11 Sequences, Series, and Power Series Sequences Many concepts in calculus involve lists of numbers that result from applying a process in stages. For example, if we use Newton’s method (Section 4.8) to approximate the zero of a function, we generate a list or sequence of numbers. If we compute average rates of change of a function over smaller and smaller intervals in order to approximate an instan- taneous rate of change (as in Section 2.7), we also generate a sequence of numbers. In the fifth century bc the Greek philosopher Zeno of Elea posed four problems, now known as Zeno’s paradoxes, that were intended to challenge some of the ideas concern- ing space and time that were held in his day. In one of his paradoxes, Zeno argued that a man standing in a room could never walk to a wall because he would first have to walk half the distance to the wall, then half the remaining distance, and then again half of what still remains, continuing in this way indefinitely (see Figure 1). The distances that the man walks at each stage form a sequence: 1 2 , 1 4 , 1 8 , 1 16 , 1 32 , . . . , 1 2 n , . . . 1 2 1 4 1 2 n . . . . . . ■ Infinite Sequences An infinite sequence, or just a sequence, can be thought of as a list of numbers written in a definite order: a 1 , a 2 , a 3 , a 4 , . . . , a n , . . . The number a 1 is called the first term, a 2 is the second term, and in general a n is the n th term. We will deal exclusively with infinite sequences and so each term a n will have a successor a n11 .
eBook - PDF- James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
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. 762 CHAPTER 11 Sequences, Series, and Power Series Sequences Many concepts in calculus involve lists of numbers that result from applying a process in stages. For example, if we use Newton’s method (Section 3.8) to approximate the zero of a function, we generate a list or sequence of numbers. If we compute average rates of change of a function over smaller and smaller intervals in order to approximate an instan- taneous rate of change (as in Section 2.1), we also generate a sequence of numbers. In the fifth century bc the Greek philosopher Zeno of Elea posed four problems, now known as Zeno’s paradoxes, that were intended to challenge some of the ideas concern- ing space and time that were held in his day. In one of his paradoxes, Zeno argued that a man standing in a room could never walk to a wall because he would first have to walk half the distance to the wall, then half the remaining distance, and then again half of what still remains, continuing in this way indefinitely (see Figure 1). The distances that the man walks at each stage form a sequence: 1 2 , 1 4 , 1 8 , 1 16 , 1 32 , . . . , 1 2 n , . . . 1 2 1 4 1 2 n . . . . . . ■ Infinite Sequences An infinite sequence, or just a sequence, can be thought of as a list of numbers written in a definite order: a 1 , a 2 , a 3 , a 4 , . . . , a n , . . . The number a 1 is called the first term, a 2 is the second term, and in general a n is the n th term. We will deal exclusively with infinite sequences and so each term a n will have a successor a n11 .
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