{"id":246009,"date":"2025-07-06T17:17:23","date_gmt":"2025-07-06T17:17:23","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=246009"},"modified":"2025-07-06T17:17:25","modified_gmt":"2025-07-06T17:17:25","slug":"matlab-required","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/06\/matlab-required\/","title":{"rendered":"MATLAB required"},"content":{"rendered":"\n

MATLAB required: Write a While loop (only the while loop) to compute Euler’s number otherwise known as the natural log e. This is a convergence problem. Euler’s number is expressed as e = \\sum_{n=0}^{\\infty} \\frac{1}{n!} This must not go to infinite so stop the loop when the term becomes < .00000001. Think about the ideas in Taylor Series. Note n! gets large very fast.<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

The formula for Euler’s number (e) is given by the Taylor series expansion:e=\u2211n=0\u221e1n!e = \\sum_{n=0}^{\\infty} \\frac{1}{n!}e=n=0\u2211\u221e\u200bn!1\u200b<\/p>\n\n\n\n

In MATLAB, we can compute Euler’s number using a while loop. The loop will stop when the term becomes less than 10\u2212810^{-8}10\u22128, or 0.000000010.000000010.00000001, as specified.<\/p>\n\n\n\n

Here\u2019s how you can write the while loop:<\/p>\n\n\n\n

matlabCopyEditn = 0;  % Initialize the counter\nterm = 1;  % The first term (1\/0!) is 1\ne_approx = 0;  % Variable to hold the approximation of e\n\nwhile term > 1e-8  % Continue until the term is less than 0.00000001\n    e_approx = e_approx + term;  % Add the current term to the sum\n    n = n + 1;  % Increment the counter\n    term = 1 \/ factorial(n);  % Calculate the next term in the series\nend\n\ndisp(e_approx)  % Display the computed value of e\n<\/code><\/pre>\n\n\n\n
Explanation:<\/h3>\n\n\n\n
    \n
  1. Initialization:<\/strong>\n
      \n
    • n<\/code> starts at 0 because the series begins with 10!\\frac{1}{0!}0!1\u200b, which is 1.<\/li>\n\n\n\n
    • term<\/code> is initialized to 1, the first term in the Taylor series expansion.<\/li>\n\n\n\n
    • e_approx<\/code> starts at 0 and will accumulate the sum of terms.<\/li>\n<\/ul>\n<\/li>\n\n\n\n
    • While loop condition:<\/strong>\n
        \n
      • The loop runs as long as term<\/code> is greater than 10\u2212810^{-8}10\u22128, ensuring we keep adding significant terms and stop when they become negligible.<\/li>\n<\/ul>\n<\/li>\n\n\n\n
      • Accumulation:<\/strong>\n
          \n
        • Inside the loop, the current term is added to e_approx<\/code>, which gradually converges to the value of eee.<\/li>\n\n\n\n
        • After each iteration, the counter n<\/code> is incremented, and term<\/code> is recalculated as 1n!\\frac{1}{n!}n!1\u200b.<\/li>\n<\/ul>\n<\/li>\n\n\n\n
        • Stopping criterion:<\/strong>\n
            \n
          • The loop stops once the term being added becomes smaller than 10\u2212810^{-8}10\u22128, which ensures the desired precision.<\/li>\n<\/ul>\n<\/li>\n\n\n\n
          • Factorial growth:<\/strong>\n
              \n
            • Factorial grows extremely fast, which is why the terms become very small quickly. This allows us to stop the computation early without needing to sum an infinite number of terms.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n
              Running this loop will give an approximation of eee that converges to the true value with good precision.<\/p>\n\n\n\n
              \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"
              MATLAB required: Write a While loop (only the while loop) to compute Euler’s number otherwise known as the natural log e. This is a convergence problem. Euler’s number is expressed as e = \\sum_{n=0}^{\\infty} \\frac{1}{n!} This must not go to infinite so stop the loop when the term becomes < .00000001. Think about the ideas […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-246009","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/246009","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=246009"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/246009\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=246009"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=246009"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=246009"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}