{"id":13375,"date":"2023-05-01T10:00:35","date_gmt":"2023-05-01T04:30:35","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=13375"},"modified":"2023-05-02T10:19:06","modified_gmt":"2023-05-02T04:49:06","slug":"exponential-series","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/exponential-series\/","title":{"rendered":"Exponential Series"},"content":{"rendered":"<h2>Exponential Series<\/h2>\n<p><iframe loading=\"lazy\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/_-x90wGBD8U?start=181&#038;feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<h3>Exponential Series<\/h3>\n<p><strong>Definition (The number e)<\/strong><br \/>\nThe limiting value of \\({ \\left( 1+\\frac { 1 }{ n } \u00a0\\right) \u00a0}^{ n }\\) when n tends to infinity is denoted by e.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128871\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-1.jpg\" alt=\"Exponential Series 1\" width=\"461\" height=\"95\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-1.jpg 461w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-1-300x62.jpg 300w\" sizes=\"(max-width: 461px) 100vw, 461px\" \/><\/p>\n<h3>Properties of e<\/h3>\n<p>(1) e lies between 2.7 and 2.8. i.e., 2.7 &lt; e &lt; 2.8.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128872\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-2.jpg\" alt=\"Exponential Series 2\" width=\"244\" height=\"58\" \/><br \/>\n(2) The value of e correct to 10 places of decimals is 2.7182818284.<br \/>\n(3) e is an irrational (incommensurable) number.<br \/>\n(4) e is the base of natural logarithm (Napier logarithm) i.e., ln x = log<sub>e<\/sub> x and log<sub>10<\/sub> e is known as Napierian constant.\u00a0log<sub>10<\/sub> e = 0.43429448, ln x = 2.303 log<sub>10<\/sub> x<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128873\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-3.jpg\" alt=\"Exponential Series 3\" width=\"580\" height=\"72\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-3.jpg 580w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-3-300x37.jpg 300w\" sizes=\"(max-width: 580px) 100vw, 580px\" \/><\/p>\n<h3>Expansion of exponential series<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128874\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-4.jpg\" alt=\"Exponential Series 4\" width=\"747\" height=\"142\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-4.jpg 747w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-4-300x57.jpg 300w\" sizes=\"(max-width: 747px) 100vw, 747px\" \/><br \/>\nThe above series known as exponential series and is called exponential function. Exponential function is also denoted by exp. i.e., e<sup>x <\/sup>\u00a0exp A = e<sup>A<\/sup>; \u2234 exp x = e<sup>x<\/sup>.<br \/>\nReplacing x by -x, we obtain<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128876\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-5.jpg\" alt=\"Exponential Series 5\" width=\"514\" height=\"191\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-5.jpg 514w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-5-300x111.jpg 300w\" sizes=\"(max-width: 514px) 100vw, 514px\" \/><\/p>\n<h3>Exponential function a<sup>x<\/sup>, where a &gt; 0<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128877\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-6.jpg\" alt=\"Exponential Series 6\" width=\"515\" height=\"320\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-6.jpg 515w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-6-300x186.jpg 300w\" sizes=\"(max-width: 515px) 100vw, 515px\" \/><\/p>\n<h3>Some standard results from exponential series<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128878\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-7.jpg\" alt=\"Exponential Series 7\" width=\"494\" height=\"661\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-7.jpg 494w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-7-224x300.jpg 224w\" sizes=\"(max-width: 494px) 100vw, 494px\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128879\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-8.jpg\" alt=\"Exponential Series 8\" width=\"595\" height=\"116\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-8.jpg 595w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Exponential-Series-8-300x58.jpg 300w\" sizes=\"(max-width: 595px) 100vw, 595px\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Exponential Series Exponential Series Definition (The number e) The limiting value of when n tends to infinity is denoted by e. Properties of e (1) e lies between 2.7 and 2.8. i.e., 2.7 &lt; e &lt; 2.8. (2) The value of e correct to 10 places of decimals is 2.7182818284. (3) e is an irrational [&hellip;]<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[5],"tags":[4868,4865,4866,4867,4869],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v17.8) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Exponential Series - A Plus Topper<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.aplustopper.com\/exponential-series\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Exponential Series\" \/>\n<meta property=\"og:description\" content=\"Exponential Series Exponential Series Definition (The number e) The limiting value of when n tends to infinity is denoted by e. 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