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Tentukan apakah deret berikut merupakan deret yang konvergen atau divergen. Berikan penjelasan Saudara.

sum_(n=1)^(oo)((-1)^(n)n)/(n^(2)+3n+5)

Tentukan apakah deret berikut merupakan deret yang konvergen atau divergen. Berikan penjelasan Saudara. $\newline$ $\sum_{n=1}^{\infty} \frac{(-1)^{n} n}{n^{2}+3 n+5}$

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Sum of finite series not start from 1

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Q. Tentukan apakah deret berikut merupakan deret yang konvergen atau divergen. Berikan penjelasan Saudara.

\newline

\sum_{n=1}^{\infty} \frac{(-1)^{n} n}{n^{2}+3 n+5}

Check Conditions for Convergence: Use the Alternating Series Test to determine convergence. $\newline$ An alternating series $\sum((-1)^n \cdot a_n)$ converges if the following conditions are met: $\newline$ $1$ . $a_{n+1} \leq a_n$ for all $n$ $\newline$ $2$ . $\lim_{n\to\infty} a_n = 0$
Compare $a_n$ and $a_{n+1}$ : Check the first condition by comparing $a_n$ and $a_{n+1}$ .
$a_n = \frac{n}{n^2 + 3n + 5}$
$a_{n+1} = \frac{n+1}{(n+1)^2 + 3(n+1) + 5}$
We need to show that $a_{n+1} \leq a_n$ for all $n$ .
Simplify $a_{n+1}$ : Simplify $a_{n+1}$ . $a_{n+1} = \frac{n+1}{n^2 + 2n + 1 + 3n + 3 + 5} = \frac{n+1}{n^2 + 5n + 9}$
Comparison of $a_n$ and $a_{n+1}$ : Compare $a_n$ and $a_{n+1}$ . For $n \geq 1$ , the denominator of $a_{n+1}$ is larger than the denominator of $a_n$ , so $a_{n+1} \leq a_n$ .
Find Limit of $a_n$ : Check the second condition by finding the limit of $a_n$ as $n$ approaches infinity. $\newline$ $\lim_{n\to\infty} \frac{n}{n^2 + 3n + 5}$ $\newline$ Use L'Hôpital's Rule or observe that the degree of the polynomial in the denominator is higher than the numerator, so the limit is $0$ .
Series Convergence: Since both conditions of the Alternating Series Test are met, the series converges.

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