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$Determine whether the sequence converges or diverges. If it converges, give the limit. a. {(5n+2)/(3n)} b. {((-1)^(n-1)(5n+2))/(3n)}$

$3$ . Determine whether the sequence converges or diverges. If it converges, give the limit. $\newline$ a. $\left\{\frac{5 n+2}{3 n}\right\}$ $\newline$ b. $\left\{\frac{(-1)^{n-1}(5 n+2)}{3 n}\right\}$

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Math Problems

Calculus

Determine end behavior of polynomial and rational functions

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3

. Determine whether the sequence converges or diverges. If it converges, give the limit.

\newline

\left\{\frac{5 n+2}{3 n}\right\}

\newline

\left\{\frac{(-1)^{n-1}(5 n+2)}{3 n}\right\}

Analysis of Sequence a: To determine whether the sequence converges or diverges and to find the limit if it converges, we will analyze each sequence separately. Let's start with sequence a, which is $\{\frac{5n+2}{3n}\}$ . We can find the limit of the sequence as $n$ approaches infinity by dividing the numerator and the denominator by $n$ , the highest power of $n$ in the denominator.
Limit of Sequence $a$ : For sequence $a$ , we have: $\newline$ $\lim_{n\to\infty} \frac{5n+2}{3n}$ $\newline$ $= \lim_{n\to\infty} \frac{5 + \frac{2}{n}}{3}$ $\newline$ Since $\frac{2}{n}$ approaches $0$ as $n$ approaches infinity, we can simplify the expression to: $\newline$ $= \lim_{n\to\infty} \frac{5}{3}$ $\newline$ $= \frac{5}{3}$ $\newline$ The limit of sequence $a$ is $\frac{5}{3}$ , which means sequence $a$ converges to $\frac{5}{3}$ .
Analysis of Sequence b: Now let's consider sequence b, which is ${((-1)^{(n-1)}(5n+2))/(3n)}$ . We can see that the sequence has an alternating sign because of the factor $(-1)^{(n-1)}$ . This means the sequence will alternate between positive and negative values.
Convergence of Sequence $b$ : To determine if sequence $b$ converges, we need to check if the absolute value of the sequence converges. The absolute value of sequence $b$ is the same as sequence $a$ , which we have already determined converges to $\frac{5}{3}$ .
Convergence of Sequence $b$ : To determine if sequence $b$ converges, we need to check if the absolute value of the sequence converges. The absolute value of sequence $b$ is the same as sequence $a$ , which we have already determined converges to $\frac{5}{3}$ .However, because the original sequence $b$ has an alternating sign, it does not converge to a single value. Instead, it oscillates between positive and negative values around the limit of its absolute value. $\newline$ Therefore, sequence $b$ diverges.