Proving the Alternating Series Test (Theorem 2.7.7) amountsto showing that the sequence of partial sums sn = a1 − a2 + a3 −· · ·±an converges. (The opening example in Section 2.1 includes a typical illustration of (sn).) Different characterizations of completeness lead to different proofs. (a) Prove the Alternating Series Test by showing that (sn) is a Cauchysequence. (b) Supply another proof for this result using the Nested Interval Property(Theorem 1.4.1). (c) Consider the subsequences (s2n) and (s2n+1), and show how the Monotone Convergence Theorem leads to a third proof for the Alternating Series Test.
Proving the Alternating Series Test (Theorem 2.7.7) amountsto showing that the sequence of partial sums sn = a1 − a2 + a3 −· · ·±an converges. (The opening example in Section 2.1 includes a typical illustration of (sn).) Different characterizations of completeness lead to different proofs. (a) Prove the Alternating Series Test by showing that (sn) is a Cauchysequence. (b) Supply another proof for this result using the Nested Interval Property(Theorem 1.4.1). (c) Consider the subsequences (s2n) and (s2n+1), and show how the Monotone Convergence Theorem leads to a third proof for the Alternating Series Test.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 68E
Related questions
Question
Proving the Alternating Series Test (Theorem 2.7.7) amounts
to showing that the sequence of partial sums
sn = a1 − a2 + a3 −· · ·±an
converges. (The opening example in Section 2.1 includes a typical illustration of (sn).) Different characterizations of completeness lead to different proofs.
(a) Prove the Alternating Series Test by showing that (sn) is a Cauchy
sequence.
(b) Supply another proof for this result using the Nested Interval Property
(Theorem 1.4.1).
(c) Consider the subsequences (s2n) and (s2n+1), and show how the Monotone Convergence Theorem leads to a third proof for the Alternating Series Test.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 6 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning