Considerthe powirSeriesっ (スキを)^n65^. 2013Find theradius ofConocrgen ccewhat isthe interval ofConucrgence(エへ interual otati on)notation)

Answered: Consider Series (スキを)^ つ。 2013 Find the…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.4: Fractional Expressions

Problem 78E

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Question

Consider
the powir
Series
っ (スキを)^
n6
5^. 2013
Find the
radius of
Conocrgen cce
what is
the interval of
Conucrgence
(エへ interual otati on)
notation)

Expert Solution

Step 1

Consider the given power series and use the ratio test to find the radius of convergence.

$\begin{matrix} a_{n} = \frac{n^{6} {(x + 8)}^{n}}{5^{n} \cdot n^{\frac{20}{3}}} \\ \lim_{n \to \infty} |\frac{a_{n + 1}}{a_{n}}| = \lim_{n \to \infty} |\frac{\frac{{(n + 1)}^{6} {(x + 8)}^{n + 1}}{5^{n + 1} \cdot {(n + 1)}^{\frac{20}{3}}}}{\frac{n^{6} {(x + 8)}^{n}}{5^{n} \cdot n^{\frac{20}{3}}}}| \\ = \lim_{n \to \infty} |\frac{{(n + 1)}^{6} {(x + 8)}^{n + 1}}{5^{n + 1} \cdot {(n + 1)}^{\frac{20}{3}}} \times \frac{5^{n} \cdot n^{\frac{20}{3}}}{n^{6} {(x + 8)}^{n}}| \\ = \lim_{n \to \infty} |\frac{{(n + 1)}^{6} {(x + 8)}^{n + 1}}{n^{6} {(x + 8)}^{n}} \times \frac{5^{n} \cdot n^{\frac{20}{3}}}{5^{n + 1} \cdot {(n + 1)}^{\frac{20}{3}}}| \\ = \lim_{n \to \infty} |{(\frac{n + 1}{n})}^{6} \times {(\frac{n}{n + 1})}^{\frac{20}{3}} \times (\frac{x + 8}{5})| \\ = \lim_{n \to \infty} |\times {(\frac{n}{n + 1})}^{\frac{2}{3}} \times (\frac{x + 8}{5})| \\ = \lim_{n \to \infty} |\times {(\frac{n}{n + 1})}^{\frac{2}{3}}| \\ = |\frac{x + 8}{5}| \times \lim_{n \to \infty} {(\frac{n}{n + 1})}^{\frac{2}{3}} \\ = |\frac{x + 8}{5}| \times \lim_{n \to \infty} {(\frac{\frac{n}{n}}{\frac{n + 1}{n}})}^{\frac{2}{3}} \\ = |\frac{x + 8}{5}| \times \lim_{n \to \infty} {(\frac{\frac{n}{n}}{\frac{n + 1}{n}})}^{\frac{2}{3}} \\ = |\frac{x + 8}{5}| \times {(\frac{1}{1 + \frac{1}{\infty}})}^{\frac{2}{3}} \\ = |\frac{x + 8}{5}| \times {(\frac{1}{1 + 0})}^{\frac{2}{3}} \\ = |\frac{x + 8}{5}| \times 1 \\ \lim_{n \to \infty} |\frac{\frac{{(n + 1)}^{6} {(x + 8)}^{n + 1}}{5^{n + 1} \cdot {(n + 1)}^{\frac{20}{3}}}}{\frac{n^{6} {(x + 8)}^{n}}{5^{n} \cdot n^{\frac{20}{3}}}}| = |\frac{x + 8}{5}| \end{matrix}$

Step 2

Further, calculate.

$\begin{array}{rcl} F o r r a d u s o f i n t e r v a l t h e r a t i o t e s t s h o u l b e l e s s t h a n 1 . \\ S o, \lim_{n \to \infty} |\frac{\frac{{(n + 1)}^{6} {(x + 8)}^{n + 1}}{5^{n + 1} \cdot {(n + 1)}^{\frac{20}{3}}}}{\frac{n^{6} {(x + 8)}^{n}}{5^{n} \cdot n^{\frac{20}{3}}}}| & < & 1 \\ |\frac{x + 8}{5}| & < & 1 \\ \frac{|x + 8|}{5} & < & 1 \\ |x + 8| & < & 5 \\ H e n c e, r a d i u s o f c o n v e r g e n c e i s 5 . \end{array}$

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Consider Series (スキを)^ つ。 2013 Find the radius of Conocrgence what is the interval of ConuLegence エへinkrual otati on)