1. Working from definitions directly, prove that a Cauchy sequence with a convergent subsequence must be convergent. 2. Prove that if we have two series of positive terms and we know that one is convergent and that each term of the other is bounded by a constant multiple of the corresponding term of the first, then the other series is also convergent. 3. There is a simpler version of the Ratio Test for series with positive terms that will be sufficient for our purposes. Namely we assume also that the ratios of successive terms form a convergent sequence. The Ratio Test for such series then says that if the limit of this sequence is less than 1 then the series is convergent whereas if the limit is greater than 1 then the series is divergent. If the limit is 1 then the Ratio Test is inconclusive. Prove this using the fact that geometric series with positive terms and common ratio less than 1 are convergent. [If you find this too hard, try to make progress with some examples.]

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 67E
icon
Related questions
Question
1. Working from definitions directly, prove that a Cauchy sequence with a convergent subsequence must be convergent.
2. Prove that if we have two series of positive terms and we know that one is convergent and that each term of the other is
bounded by a constant multiple of the corresponding term of the first, then the other series is also convergent.
3. There is a simpler version of the Ratio Test for series with positive terms that will be sufficient for our purposes. Namely we
assume also that the ratios of successive terms form a convergent sequence. The Ratio Test for such series then says that if the
limit of this sequence is less than 1 then the series is convergent whereas if the limit is greater than 1 then the series is
divergent. If the limit is 1 then the Ratio Test is inconclusive. Prove this using the fact that geometric series with positive terms
and common ratio less than 1 are convergent. [If you find this too hard, try to make progress with some examples.]
Transcribed Image Text:1. Working from definitions directly, prove that a Cauchy sequence with a convergent subsequence must be convergent. 2. Prove that if we have two series of positive terms and we know that one is convergent and that each term of the other is bounded by a constant multiple of the corresponding term of the first, then the other series is also convergent. 3. There is a simpler version of the Ratio Test for series with positive terms that will be sufficient for our purposes. Namely we assume also that the ratios of successive terms form a convergent sequence. The Ratio Test for such series then says that if the limit of this sequence is less than 1 then the series is convergent whereas if the limit is greater than 1 then the series is divergent. If the limit is 1 then the Ratio Test is inconclusive. Prove this using the fact that geometric series with positive terms and common ratio less than 1 are convergent. [If you find this too hard, try to make progress with some examples.]
Expert Solution
steps

Step by step

Solved in 2 steps

Blurred answer
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra
College Algebra
Algebra
ISBN:
9781938168383
Author:
Jay Abramson
Publisher:
OpenStax