{"id":9444,"date":"2022-07-19T14:00:48","date_gmt":"2022-07-19T11:00:48","guid":{"rendered":"https:\/\/starlanguageblog.com\/?p=9444"},"modified":"2022-07-19T14:00:48","modified_gmt":"2022-07-19T11:00:48","slug":"the-alternating-series-error-bound","status":"publish","type":"post","link":"https:\/\/www.starlanguageblog.com\/the-alternating-series-error-bound\/","title":{"rendered":"The Alternating Series Error Bound"},"content":{"rendered":"<h1><span style=\"font-weight: 400;\">The Alternating Series Error Bound<\/span><\/h1>\n<p>Alternating Series Error Bound: Given an alternating series \u2211\u221en=0an \u2211 n = 0 \u221e a n , the alternating series error bound is given by: \u2223\u2223\u2211\u221en=0an\u2212\u2211kn=0an\u2223\u2223&lt;|an+1| | \u2211 n = 0 \u221e a n \u2212 \u2211 n = 0 k a n | &lt; | a n + 1 | .<\/p>\n<p><span style=\"font-weight: 400;\">The alternating series error bound is a mathematical function that guarantees a <a href=\"https:\/\/starlanguageblog.com\/10-words-to-describe-the-strength-of-a-godly-woman-10-special-qualities\/\">minimum<\/a> value, n, for a test. In other words, it <a href=\"https:\/\/starlanguageblog.com\/what-are-some-ideas-and-examples-of-physical-touch-love-language-long-distance-for-her\/\">guarantees<\/a> that the error is at most 0.1. In practice, this will be the same value as 0.1, which is why the error bound is also known as the Lagrange error. In the following paragraphs, we will discuss the error bound and how it can be calculated.<\/span><\/p>\n<h2><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone  wp-image-9446\" src=\"https:\/\/wordpress-1162154-4054340.cloudwaysapps.com\/wp-content\/uploads\/2022\/07\/ezgif.com-gif-maker-78-1.webp\" alt=\"The Alternating Series Error Bound\" width=\"641\" height=\"431\" \/><\/h2>\n<h2><span style=\"font-weight: 400;\">Lagrange error bound<\/span><\/h2>\n<p><span style=\"font-weight: 400;\">The Lagrange error bound estimates the residual value after a term in alternating series. The remainder will lie in an interval around the actual value. Then, a partial sum of the series will jump around the value. Consequently, the <a href=\"https:\/\/starlanguageblog.com\/elementary-school-grades-and-ages\/\">error<\/a> bound will be smaller than the value. The smaller the error bound, the better. Hence, a minor error bound can be used in alternating series.<\/span><\/p>\n<h2><span style=\"font-weight: 400;\">Upper bound computations on the error<\/span><\/h2>\n<p><span style=\"font-weight: 400;\">In the context of alternating series error bounds, the upper bound on f is a measure of the error. It may be close to the actual error or off by a significant amount. In the following section, we will discuss upper bound computations for alternating series error bounds. It will also be clear that upper bound computations of f are a good starting point. First, it is essential to understand the assumptions involved before performing upper-bound computations.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">To calculate the upper bound, use the alternating series as the basis. The alternating series is a sequence of terms that alternate in sign and magnitude. The terms in an alternating series are either smaller or larger than their predecessors. In both cases, the sequence is said to converge if the alternating series is infinite. Usually, a series converges at some point. Therefore, upper bound computations on the alternating series error bounds are practical when estimating uncertainty in a series.<\/span><\/p>\n<h2><span style=\"font-weight: 400;\">Upper bound computations on the actual error<\/span><\/h2>\n<p><span style=\"font-weight: 400;\">It is possible to compute upper bounds on the actual error of an alternating series. These bounds can be very close to the actual error, but they may also be quite off. For example, when a given series has an error of 0.5, the upper <a href=\"https:\/\/starlanguageblog.com\/how-to-write-a-letter-please-excuse-my-daughters-absent-school\/\">bound<\/a> would be f(n+1)(z), which would be about eight times the actual error. This result is also known as the divergence test.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">An alternating series can be computed to three decimal places using the sum of its terms. To determine the error bound, we use the first term and its corresponding error. If we ignore this term, the alternating series is infinity. The associated sequence of partial sums converges, and the alternating <a href=\"https:\/\/starlanguageblog.com\/this-is-he-or-this-is-him-which-is-the-correct-form-of-english\/\">series<\/a> has an error of less than the first term. Moreover, a negative sign indicates that the series is not stable, so the upper bound is not a valid bound.<\/span><\/p>\n<h2><span style=\"font-weight: 400;\">Guarantees of the error bound.<\/span><\/h2>\n<p><span style=\"font-weight: 400;\">A partial sum n is the maximum number of terms that appear in the alternating series, and the number of terms may vary between odd and even. The alternating series is a partial sum whose terms alternate above and below the final limit. Understanding this property leads to a partial sum error bound. It is possible to approximate the value of n by the partial sum Sk. The alternating series error bound is a simple way to determine the error rate of a partial sum.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">This property of the alternating series error bound provides a <a href=\"https:\/\/starlanguageblog.com\/what-language-did-adam-and-eve-speak-islam-what-about-abraham\/\">deterministic<\/a> upper bound for the error. This means that the error rate is as close to the actual value as possible without assuming that the number of terms is infinite. Despite this property, many upper-bound error computations are off quite a bit. The alternating series error bound is not a foolproof formula, however. In practice, it is a valuable tool to use in many scientific applications.<\/span><\/p>\n<h2><span style=\"font-weight: 400;\">Guarantees of the error bound for all integers k&gt;=0<\/span><\/h2>\n<p><span style=\"font-weight: 400;\">Suppose a function f is a sequence of finite terms and k is the first natural number in the series. If k is a positive integer, then the nth degree Taylor polynomial is given by C(-1)&#8221;_. If k is an integer, then k2 + k+1 will also be a positive number.<\/span><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The Alternating Series Error Bound Alternating Series Error Bound: Given an alternating series \u2211\u221en=0an \u2211 n = 0 \u221e a n , the alternating series error bound is given by: \u2223\u2223\u2211\u221en=0an\u2212\u2211kn=0an\u2223\u2223&lt;|an+1| | \u2211 n = 0 \u221e a n \u2212 \u2211 n = 0 k a n | &lt; | a n + 1 | [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":9445,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[81],"tags":[620,621],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v22.3 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>The Alternating Series Error Bound - Star Language Blog<\/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.starlanguageblog.com\/the-alternating-series-error-bound\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The Alternating Series Error Bound - Star Language Blog\" \/>\n<meta property=\"og:description\" content=\"The Alternating Series Error Bound Alternating Series Error Bound: Given an alternating series \u2211\u221en=0an \u2211 n = 0 \u221e a n , the alternating series error bound is given by: \u2223\u2223\u2211\u221en=0an\u2212\u2211kn=0an\u2223\u2223&lt;|an+1| | \u2211 n = 0 \u221e a n \u2212 \u2211 n = 0 k a n | &lt; | a n + 1 | [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.starlanguageblog.com\/the-alternating-series-error-bound\/\" \/>\n<meta property=\"og:site_name\" content=\"Star Language Blog\" \/>\n<meta property=\"article:published_time\" content=\"2022-07-19T11:00:48+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.starlanguageblog.com\/wp-content\/uploads\/2022\/07\/ezgif.com-gif-maker-77.webp\" \/>\n\t<meta property=\"og:image:width\" content=\"615\" \/>\n\t<meta property=\"og:image:height\" content=\"412\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/webp\" \/>\n<meta name=\"author\" content=\"Starla\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Starla\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/www.starlanguageblog.com\/the-alternating-series-error-bound\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/www.starlanguageblog.com\/the-alternating-series-error-bound\/\"},\"author\":{\"name\":\"Starla\",\"@id\":\"https:\/\/www.starlanguageblog.com\/#\/schema\/person\/18c6c447348b68f36676245bfe3f83fc\"},\"headline\":\"The Alternating Series Error Bound\",\"datePublished\":\"2022-07-19T11:00:48+00:00\",\"dateModified\":\"2022-07-19T11:00:48+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/www.starlanguageblog.com\/the-alternating-series-error-bound\/\"},\"wordCount\":728,\"publisher\":{\"@id\":\"https:\/\/www.starlanguageblog.com\/#organization\"},\"image\":{\"@id\":\"https:\/\/www.starlanguageblog.com\/the-alternating-series-error-bound\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/www.starlanguageblog.com\/wp-content\/uploads\/2022\/07\/ezgif.com-gif-maker-77.webp\",\"keywords\":[\"Series\",\"The Alternating Series Error Bound\"],\"articleSection\":[\"Computer Language\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.starlanguageblog.com\/the-alternating-series-error-bound\/\",\"url\":\"https:\/\/www.starlanguageblog.com\/the-alternating-series-error-bound\/\",\"name\":\"The Alternating Series Error Bound - 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