Alternating Series
An alternating series is a series in which the terms alternate in sign, such as +a - b + c - d. For an alternating series, the terms must decrease in absolute value and approach zero as the series progresses. The alternating series test is a method used to determine the convergence of such series.
3 Key excerpts on "Alternating Series"
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...12 Infinite Series and Integrals Due to their algebraic, analytical, and numerical simplicity, polynomials are often elected for description of (more) involved analytical functions; this is notably the case of Taylor’s series – which (when convergent) provides the exact value of the associated function, should the number of terms tend to infinity. Related concepts encompass Euler’s infinite product, as well as Euler’s original definition of factorial of an integer number – a concept that has been extended to any real number, at the expense of gamma (integral) function. 12.1 Definition and Criteria of Convergence Recalling the concept of infinite series, it is more or less intuitive that cannot converge unless. In fact, if the n ‐term series, S n, is considered, viz. (12.1) as well as the corresponding (n − 1)‐term series, S n− 1, i.e. (12.2) then ordered subtraction of Eq. (12.2) from Eq. (12.1) unfolds (12.3) if the series converges to sum S as per Eq. (2.73), then (12.4) which readily implies (12.5) at the expense of Eq. (9.73). On the other hand, application of limits to both sides of Eq. (12.3) gives rise to (12.6) so elimination of between Eqs. (12.5) and (12.6) finally yields (12.7) Equation (12.7) is a necessary, but not sufficient condition for convergence; for instance, the harmonic series – defined as, leads to an infinite sum, even though (12.8) after setting a n ≡ in Eq. (12.7), and recalling Eq. (9.108). When a series consists of only positive terms (i.e. a i ≥ 0 for all i), then it must either converge or diverge to +∞ ; it clearly cannot oscillate, as might happen when both positive and negative terms are present...
- J. Rosebush, Stu Schwartz(Authors)
- 2016(Publication Date)
- Research & Education Association(Publisher)
...If it converges, its sum is given by. 2. p - series,, converges when p > 1 and diverges when 0 < p ≤ 1. 3. Alternating Series are series with terms whose signs alternate. They are of the form. 4. Harmonic series,, diverges. This is a p -series with p = 1. 5. Alternating Harmonic series,, converges. 6. Alternating p - series converges for p > 0. 7. Power series in x, Power series in (x – a),. (More on power series later on.) 8. Telescoping series is a series in which all but a finite number of terms cancel out. It is either decomposed into partial fractions or you need to decompose it yourself. For example,. This series converges because. Just because a fraction can be decomposed into its partial fractions, does not mean it will be telescoping! Not all telescoping series converge. For example: and, so the series diverges. B. Convergence/Divergence Tests for Series Let be an infinite series of positive terms. The series converges if and only if the sequence of partial sums,, converges. Also,. That is, the sum of the series equals the limit of the sequence of partial sums. Also, if a series converges absolutely, then it converges. This means that if converges, then converges. For example, converges because, or, equivalently, converges (p -series with p > 1). 1. Divergence Test If, the series diverges. The contrapositive of this statement, which is logically equivalent to the statement, is also very useful. That is, if converges, then. In other words, if a series is convergent, its terms must approach zero. However, does not imply convergence. i. The series is divergent since ii. An example of a series in which the terms approach zero but which is not convergent is the harmonic series,. 2. Ratio Test (a) If then the series converges; (b) if the series diverges. If, this test is inconclusive; use a different convergence test. Specifically, the Ratio Test does not work for p -series because in that case,...
- Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...STUDY UNIT V NUMBER PATTERNS AND CALCULUS PREVIEW 15 SEQUENCES, SERIES, AND COUNTING WHAT YOU WILL LEARN Lists of numbers sometimes follow predictable patterns. Arithmetic and geometric sequences have the property that each term in a sequence after the first is obtained by adding or multiplying the preceding term by the same number. Some number patterns that involve only nonnegative integers can be generalized by statements that look like algebraic formulas. These formulas are proved using a special method called mathematical induction. Expanding a binomial such as (a + b) 2 is not difficult. However, when the exponent of the binomial is greater than 2, a good deal of effort is required to expand the binomial using repeated multiplication. The binomial theorem provides an algebraic formula that tells how to find each term in the expansion of a binomial of the form (a + b) n, where n is a positive integer. LESSONS IN CHAPTER 15 • Lesson 15-1: Arithmetic Sequences and Series • Lesson 15-2: Geometric Sequences and Series • Lesson 15-3: Generalized Sequences • Lesson 15-4: Mathematical Induction • Lesson 15-5: Permutations and Combinations • Lesson 15-6: The Binomial Theorem Lesson 15-1: Arithmetic Sequences and Series KEY IDEAS A sequence is a list of numbers, called terms, written in a specific order. A list of numbers such as 2, 5, 8, 11, 14, … is called an arithmetic sequence since each term after the first is obtained by adding a constant, 3 in this case, to the term that precedes it. A sequence of numbers may be finite or infinite. A finite sequence has a definite number of terms. An infinite sequence is nonending. An infinite sequence uses three trailing periods to indicate that the pattern never ends, as in 2, 5, 8, 11, 14, …. COMMON DIFFERENCE In an arithmetic sequence, subtracting any term from the term that follows it always results in the same number. This number is called the common difference and is denoted as d...
