{"id":193790,"date":"2024-07-31T02:24:48","date_gmt":"2024-07-31T07:24:48","guid":{"rendered":"https:\/\/lifeboat.com\/blog\/2024\/07\/concerning-an-infinite-series-of-ramanujan-related-to-the-natural-logarithm"},"modified":"2024-07-31T02:24:48","modified_gmt":"2024-07-31T07:24:48","slug":"concerning-an-infinite-series-of-ramanujan-related-to-the-natural-logarithm","status":"publish","type":"post","link":"https:\/\/lifeboat.com\/blog\/2024\/07\/concerning-an-infinite-series-of-ramanujan-related-to-the-natural-logarithm","title":{"rendered":"Concerning an infinite series of Ramanujan related to the natural logarithm"},"content":{"rendered":"<p><a class=\"aligncenter blog-photo\" href=\"https:\/\/lifeboat.com\/blog.images\/concerning-an-infinite-series-of-ramanujan-related-to-the-natural-logarithm.jpg\"><\/a><\/p>\n<p>We consider an infinite series, due to Ramanujan, which converges to a simple expression involving the natural logarithm. We show that Ramanujan\u2019s series represents a completely monotone function, and explore some of its consequences, including a non-trivial family of inequalities satisfied by the natural logarithm, some formulas for the Euler\u2013Mascheroni constant, and a recurrence satisfied by the Bernoulli numbers. We also provide a one-parameter generalization of Ramanujan\u2019s series, which includes as a special case another related infinite series evaluation due to Ramanujan.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>We consider an infinite series, due to Ramanujan, which converges to a simple expression involving the natural logarithm. We show that Ramanujan\u2019s series represents a completely monotone function, and explore some of its consequences, including a non-trivial family of inequalities satisfied by the natural logarithm, some formulas for the Euler\u2013Mascheroni constant, and a recurrence satisfied [\u2026]<\/p>\n","protected":false},"author":513,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[20],"tags":[],"class_list":["post-193790","post","type-post","status-publish","format-standard","hentry","category-futurism"],"_links":{"self":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/posts\/193790","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/users\/513"}],"replies":[{"embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/comments?post=193790"}],"version-history":[{"count":0,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/posts\/193790\/revisions"}],"wp:attachment":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/media?parent=193790"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/categories?post=193790"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/tags?post=193790"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}