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Tentukan jari-jari konvergen deret berikut dengan cara formula Cauchy Hadamard: (bobot 20)

sum_(n=2)^(oo)(3^(n))/((n-2))(z)^(n)
SELAMAT MENGERJAKAN
=

$5$ . Tentukan jari-jari konvergen deret berikut dengan cara formula Cauchy Hadamard: (bobot $20$ ) $\newline$ $\sum_{n=2}^{\infty} \frac{3^{n}}{(n-2)}(z)^{n}$ $\newline$ SELAMAT MENGERJAKAN $=$

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Sum of finite series not start from 1

Full solution

Q.

5

. Tentukan jari-jari konvergen deret berikut dengan cara formula Cauchy Hadamard: (bobot

20

)

\newline

\sum_{n=2}^{\infty} \frac{3^{n}}{(n-2)}(z)^{n}

\newline

SELAMAT MENGERJAKAN

=

Write Cauchy-Hadamard formula: Write down the Cauchy-Hadamard formula. $\newline$ The radius of convergence $R$ is given by $\frac{1}{L}$ , where $L$ is the limit superior of the nth root of the absolute value of the nth term.
Identify nth term: Identify the $n$ th term of the series. $\newline$ The $n$ th term is $a_n = \frac{3^n}{(n-2)z^n}$ .
Calculate nth root: Calculate the nth root of the absolute value of the nth term. $\newline$ We take the nth root of $|a_n|$ which is $\left|\frac{3^n}{(n-2)z^n}\right|^{\frac{1}{n}}$ .
Simplify expression: Simplify the expression. $\newline$ The $n$ th root of $|a_n|$ becomes $|\frac{3}{z}| \cdot |\frac{1}{(n-2)^{\frac{1}{n}}}|$ .
Find limit: Find the limit as $n$ approaches infinity. The limit of $|\frac{1}{(n-2)^{\frac{1}{n}}}|$ as $n$ approaches infinity is $1$ , so $L = |\frac{3}{z}|$ .
Calculate radius: Calculate the radius of convergence. $\newline$ The radius of convergence $R$ is $\frac{1}{L}$ , so $R = \frac{1}{|\frac{3}{z}|} = \frac{|z|}{3}$ .

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