{"id":217841,"date":"2025-05-22T14:41:55","date_gmt":"2025-05-22T14:41:55","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=217841"},"modified":"2025-05-22T14:41:57","modified_gmt":"2025-05-22T14:41:57","slug":"identify-the-function-represented-by-the-following-power-series-sigma","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/22\/identify-the-function-represented-by-the-following-power-series-sigma\/","title":{"rendered":"Identify the function represented by the following power series Sigma"},"content":{"rendered":"\n

Identify the function represented by the following power series Sigma (-1)kxk\/3k Click the icon to view a table of Taylor series for common functions F(x) =<\/p>\n\n\n\n

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

The power series given is:<\/p>\n\n\n\n

$$
\\sum_{k=0}^{\\infty} \\frac{(-1)^k x^k}{3^k}
$$<\/p>\n\n\n\n

This can be rewritten as:<\/p>\n\n\n\n

$$
\\sum_{k=0}^{\\infty} \\left(\\frac{-x}{3}\\right)^k
$$<\/p>\n\n\n\n

This is a geometric series with the general form:<\/p>\n\n\n\n

$$
\\sum_{k=0}^{\\infty} r^k = \\frac{1}{1 – r}, \\quad \\text{for } |r| < 1
$$<\/p>\n\n\n\n

In our case, the common ratio $r = \\frac{-x}{3}$. Thus, the series converges when:<\/p>\n\n\n\n

$$
\\left|\\frac{-x}{3}\\right| < 1 \\Rightarrow |x| < 3
$$<\/p>\n\n\n\n

Using the geometric series formula:<\/p>\n\n\n\n

$$
\\sum_{k=0}^{\\infty} \\left(\\frac{-x}{3}\\right)^k = \\frac{1}{1 – \\left(\\frac{-x}{3}\\right)} = \\frac{1}{1 + \\frac{x}{3}} = \\frac{1}{\\frac{3 + x}{3}} = \\frac{3}{3 + x}
$$<\/p>\n\n\n\n

\u2705 Final Answer:<\/h3>\n\n\n\n

$$
\\boxed{f(x) = \\frac{3}{3 + x}}
$$<\/p>\n\n\n\n

\n\n\n\n

\ud83d\udcda 300-Word Explanation:<\/h3>\n\n\n\n

The given power series is:<\/p>\n\n\n\n

$$
\\sum_{k=0}^{\\infty} \\frac{(-1)^k x^k}{3^k}
$$<\/p>\n\n\n\n

To identify the function it represents, it’s helpful to recognize its structure. Notice that the numerator alternates in sign, due to $(-1)^k$, and the denominator has a growing power of 3, that is $3^k$. Thus, we can simplify the general term:<\/p>\n\n\n\n

$$
\\frac{(-1)^k x^k}{3^k} = \\left(\\frac{-x}{3}\\right)^k
$$<\/p>\n\n\n\n

This matches the general form of a geometric series:<\/p>\n\n\n\n

$$
\\sum_{k=0}^{\\infty} r^k = \\frac{1}{1 – r}, \\quad \\text{where } |r| < 1
$$<\/p>\n\n\n\n

The value of $r$ in this case is $\\frac{-x}{3}$. According to the geometric series rule, this infinite series converges to a function when the absolute value of the common ratio is less than 1, i.e., $|x| < 3$.<\/p>\n\n\n\n

Now, we apply the formula:<\/p>\n\n\n\n

$$
\\sum_{k=0}^{\\infty} \\left(\\frac{-x}{3}\\right)^k = \\frac{1}{1 – \\left(-\\frac{x}{3}\\right)} = \\frac{1}{1 + \\frac{x}{3}}
$$<\/p>\n\n\n\n

To simplify this expression, combine the terms in the denominator:<\/p>\n\n\n\n

$$
\\frac{1}{1 + \\frac{x}{3}} = \\frac{1}{\\frac{3 + x}{3}} = \\frac{3}{3 + x}
$$<\/p>\n\n\n\n

Therefore, the original power series represents the function $\\boxed{f(x) = \\frac{3}{3 + x}}$, valid for $|x| < 3$.<\/p>\n\n\n\n

This type of identification is useful in calculus, especially in understanding how power series can represent functions within an interval of convergence.<\/p>\n\n\n\n

\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

Identify the function represented by the following power series Sigma (-1)kxk\/3k Click the icon to view a table of Taylor series for common functions F(x) = The Correct Answer and Explanation is: The power series given is: $$\\sum_{k=0}^{\\infty} \\frac{(-1)^k x^k}{3^k}$$ This can be rewritten as: $$\\sum_{k=0}^{\\infty} \\left(\\frac{-x}{3}\\right)^k$$ This is a geometric series with the general […]<\/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-217841","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217841","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=217841"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217841\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=217841"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=217841"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=217841"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}