{"id":194966,"date":"2025-02-25T10:24:05","date_gmt":"2025-02-25T10:24:05","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=194966"},"modified":"2025-02-25T10:24:08","modified_gmt":"2025-02-25T10:24:08","slug":"use-differentiation-to-find-a-power-series-representation-for-fx","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/25\/use-differentiation-to-find-a-power-series-representation-for-fx\/","title":{"rendered":"Use differentiation to find a power series representation for f(x) ="},"content":{"rendered":"\n

(a) Use differentiation to find a power series representation for f(x) =<\/p>\n\n\n\n

1<\/p>\n\n\n\n

(6 + x)2<\/p>\n\n\n\n

.<\/p>\n\n\n\n

f(x) =<\/p>\n\n\n\n

\u221e<\/p>\n\n\n\n

leftparen1.gif<\/p>\n\n\n\n

(\u22121)n(n+1)xn6n+2\u200b<\/p>\n\n\n\n

rightparen1.gif<\/p>\n\n\n\n

sum.gif<\/p>\n\n\n\n

n = 0<\/p>\n\n\n\n

What is the radius of convergence, R?
R =<\/p>\n\n\n\n

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

To find a power series representation for<\/p>\n\n\n\n

[
f(x) = \\frac{1}{(6 + x)^2}
]<\/p>\n\n\n\n

Step 1: Consider a Related Geometric Series<\/strong><\/h3>\n\n\n\n

We start with the standard geometric series representation:<\/p>\n\n\n\n

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

To apply this, rewrite ( f(x) ) in a suitable form. First, express the denominator as:<\/p>\n\n\n\n

[
\\frac{1}{6+x} = \\frac{1}{6} \\cdot \\frac{1}{1 + \\frac{x}{6}}.
]<\/p>\n\n\n\n

Using the geometric series for ( \\frac{1}{1 – (-x\/6)} ), we obtain:<\/p>\n\n\n\n

[
\\frac{1}{6 + x} = \\frac{1}{6} \\sum_{n=0}^{\\infty} \\left( -\\frac{x}{6} \\right)^n, \\quad \\text{for } |x| < 6.
]<\/p>\n\n\n\n

Step 2: Differentiate Both Sides<\/strong><\/h3>\n\n\n\n

Since we need ( \\frac{1}{(6+x)^2} ), differentiate both sides with respect to ( x ):<\/p>\n\n\n\n

[
\\frac{d}{dx} \\left[ \\frac{1}{6+x} \\right] = \\frac{d}{dx} \\left[ \\frac{1}{6} \\sum_{n=0}^{\\infty} \\left( -\\frac{x}{6} \\right)^n \\right].
]<\/p>\n\n\n\n

Using the derivative of ( \\frac{1}{6 + x} ), we get:<\/p>\n\n\n\n

[
-\\frac{1}{(6+x)^2} = \\frac{1}{6} \\sum_{n=1}^{\\infty} n \\left( -\\frac{x}{6} \\right)^{n-1} \\cdot \\left( -\\frac{1}{6} \\right).
]<\/p>\n\n\n\n

Simplifying,<\/p>\n\n\n\n

[
\\frac{1}{(6+x)^2} = \\sum_{n=1}^{\\infty} \\frac{n}{6^{n+1}} (-x)^{n-1}.
]<\/p>\n\n\n\n

Rewriting the index by setting ( n \\to n+1 ), we obtain:<\/p>\n\n\n\n

[
f(x) = \\sum_{n=0}^{\\infty} (n+1) (-1)^n \\frac{x^n}{6^{n+2}}.
]<\/p>\n\n\n\n

Step 3: Radius of Convergence<\/strong><\/h3>\n\n\n\n

The radius of convergence ( R ) is determined by the original geometric series, which converges for ( |x\/6| < 1 ), implying:<\/p>\n\n\n\n

[
|x| < 6.
]<\/p>\n\n\n\n

Thus, the radius of convergence is:<\/p>\n\n\n\n

[
R = 6.
]<\/p>\n\n\n\n

\n\n\n\n

Now, let’s generate a visual representation of the power series.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Here is a visual representation of the function ( f(x) = \\frac{1}{(6+x)^2} ) alongside its power series approximation using 10 terms. The dashed red line represents the series approximation, while the solid blue line is the exact function. The dotted vertical lines at ( x = \\pm 6 ) indicate the radius of convergence ( R = 6 ).<\/p>\n","protected":false},"excerpt":{"rendered":"

(a) Use differentiation to find a power series representation for f(x) = 1 (6 + x)2 . f(x) = \u221e leftparen1.gif (\u22121)n(n+1)xn6n+2\u200b rightparen1.gif sum.gif n = 0 What is the radius of convergence, R?R = The Correct Answer and Explanation is : To find a power series representation for [f(x) = \\frac{1}{(6 + x)^2}] Step […]<\/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-194966","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/194966","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=194966"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/194966\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=194966"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=194966"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=194966"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}