Mathematics Foundations/4.3 Sequences and Series

Sequences and Series Introduction to Sequences A sequence is an ordered list of numbers. Formally, a sequence can be defined as a function whose domain is the set of natural numbers (or a subset of the natural numbers). Definition A sequence (or simply ) is a function where is the -th term of the sequence. Examples of Sequences - The sequence of natural numbers - A sequence that approaches 0 - An oscillating sequence - A sequence that grows without bound Convergence of Sequences The concept of convergence is central to the study of sequences and forms the foundation for calculus and analysis. Definition of Convergence A sequence converges to a limit if for every , there exists a positive integer such that for all . We write or as . If a sequence does not converge, it is said to diverge. Examples does not exist (the sequence diverges) does not exist (the sequence diverges to infinity) Properties of Convergent Sequences If and , then: 1. Sum Rule: 2. Difference Rule: 3. Product Rule: 4. Quotient Rule: (provided ) 5. Constant Multiple Rule: for any constant Monotone Sequences A sequence is: Monotone increasing if for all Strictly monotone increasing if for all Monotone decreasing if for all Strictly monotone decreasing if for all Monotone Convergence Theorem Theorem: Every bounded monotone sequence converges. Specifically: If is monotone increasing and bounded above, then If is monotone decreasing and bounded below, then This theorem is a direct consequence of the completeness of the real numbers. Bounded Sequences A sequence is bounded if there exists such that for all . Bolzano-Weierstrass Theorem Theorem: Every bounded sequence has a convergent subsequence. This is another important result that follows from the completeness of the real numbers. Cauchy Sequences A sequence is a Cauchy sequence if for every , there exists a positive integer such that for all . Cauchy Criterion Theorem: A sequence of real numbers converges if and only if it is a Cauchy sequence. This equivalence between convergence and the Cauchy property is yet another manifestation of the completeness of the real numbers. Introduction to Series A series is the sum of the terms of a sequence. Definition Given a sequence , the corresponding series is denoted by and represents the infinite sum The partial sums of the series are defined as Convergence of Series A series converges to a sum if the sequence of partial sums converges to : If the sequence of partial sums does not converge, the series is said to diverge. Types of Series Geometric Series A geometric series has the form Theorem: A geometric series converges if and only if . When it converges, the sum is . Examples: Telescoping Series A telescoping series is one where many terms cancel in the partial sums, leaving only a few terms. Example: Consider The partial sum As , , so the series converges to 1. p-Series A p-series has the form Theorem: A p-series converges if and only if . Examples: (the harmonic series) diverges () converges () Tests for Convergence Comparison Test If for all (for some ), then: If converges, then converges If diverges, then diverges Limit Comparison Test If and for all (for some ), and where , then and either both converge or both diverge. Ratio Test If , then: If , the series converges absolutely If or , the series diverges If , the test is inconclusive Root Test If , then: If , the series converges absolutely If or , the series diverges If , the test is inconclusive Integral Test If is a positive, continuous, decreasing function on with for all integers , then converges if and only if converges. Absolute and Conditional Convergence A series is said to converge absolutely if converges. A series is said to converge conditionally if converges but diverges. Theorem: If a series converges absolutely, then it converges. Example: The alternating harmonic series converges conditionally but not absolutely. Exercises 1. Determine whether the sequence defined by converges, and if so, find its limit. 2. Prove that the sequence defined by and converges. 3. Determine the convergence or divergence of the series . 4. Test the convergence of . 5. Prove that if and both converge absolutely, then and also converge. 6. Determine whether the series converges absolutely, converges conditionally, or diverges.