- \n
- The Monotone Convergence Theorem: If a sequence is monotone and bounded, then it converges.<\/li>\n
- Theorem 2.5.2: Subsequences of a convergent sequence converge to the same limit as the original sequence.<\/li>\n
- Bolzano-Weierstrass Theorem: Every bounded sequence contains a convergent subsequence.<\/li>\n
- Cauchy Criterion: A sequence converges if and only if it is a Cauchy Sequence.<\/li>\n
- We showed in challenge 4 of homework 1 that if
, then
.<\/li>\n<\/ul>\n
Convergence Theorems for Series<\/strong><\/p>\n
- \n
- The Cauchy Condensation Test: Suppose
is decreasing and satisfies
for all
. Then the series
converges if and only if the series\u00a0
converges.<\/li>\n
- The Cauchy Criterion for Series: The series\u00a0
converges if and only if, given
, there exists an
such that whenever
it follows that
.<\/li>\n
- Theorem 2.7.3: If the series
.<\/li>\n
- Comparison Test: Assume
and
are sequences satisfying
for all
i) If\u00a0
converges, then\u00a0
diverges.<\/li>\n
- Absolute Convergence Test: If the series\u00a0
converges, then\u00a0
converges as well.<\/li>\n
- Alternating Series Test: Let
be a sequence satisfying (i)
and (ii)
. Then the alternating series\u00a0
converges.<\/li>\n<\/ul>\n
Note that while these theorems apply specifically to sequences or series, the convergence of the series\u00a0<\/strong>
. Thus, you can also show the convergence of a series by identifying a formula and showing convergence for its sequence of partial sums.<\/p>\n","protected":false},"excerpt":{"rendered":"
Chapter 2 introduces sequences, series, and several theorems for proving the convergence or divergence for each. However, the theorems cannot be used interchangeably for sequences and series. This Muddiest Point post will distinguish between some theorems for sequences and series. Convergence Theorems for Sequences The Monotone Convergence Theorem: If a sequence is monotone and bounded, […]<\/p>\n","protected":false},"author":3528,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[58822],"tags":[],"class_list":["post-838","post","type-post","status-publish","format-standard","hentry","category-muddiest-point"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7L4E1-dw","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/838","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/users\/3528"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=838"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/838\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=838"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=838"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=838"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
- The Cauchy Condensation Test: Suppose