Sequence

Examples

Find the general term for each sequence:

${1, \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \dots}$

\frac{1}{2 n - 1}

${5, 8, 11, 14, 17, \dots}$

3 n + 3

${- \frac{1}{2}, \frac{4}{3}, \frac{9}{4}, \frac{16}{5}, \dots}$

\frac{(- 1)^{n} n^{2}}{n + 1}

${\frac{3}{5}, - \frac{4}{25}, \frac{1}{25}, - \frac{6}{625}, \dots}$

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

${3, 2, \frac{4}{3}, \frac{8}{9}, \frac{16}{27}, \dots}$

\frac{2^{n - 1}}{3^{n - 2}}

Determine if the sequence converges or diverges:

${e^{\frac{2 n}{n + 2}} \arctan \frac{n}{2}}_{n = 1}^{\infty}$

lim_{n \to \infty} e^{\frac{2 n}{n + 2}} \arctan \frac{n}{2} = e^{2} \frac{π}{2}

${\frac{(n + 1)!}{(n + 4)!}}_{n = 1}^{\infty}$

\begin{aligned} lim_{n \to \infty} \frac{(n + 1)!}{(n + 4)!} & = lim_{n \to \infty} \frac{(n + 1)!}{(n + 4) (n + 3) (n + 2) (n + 1)!} \\ = lim_{n \to \infty} \frac{1}{(n + 4) (n + 3) (n + 2)} \\ = 0 \end{aligned}

${\frac{(n + 2)!}{n!}}_{n = 1}^{\infty}$

\begin{aligned} lim_{n \to \infty} \frac{(n + 2)!}{n!} & = lim_{n \to \infty} \frac{(n + 2) (n + 1) n!}{n!} \\ = lim_{n \to \infty} (n + 2) (n + 1) \\ = \infty \end{aligned}

${\ln (1 + n^{2}) - \frac{1}{2} \ln (n^{3} + 4 n)}_{n = 1}^{\infty}$

lim_{n \to \infty} \ln (1 + n^{2}) - \frac{1}{2} \ln (n^{3} + 4 n) = - \infty

${(1 + 2 n)^{\frac{1}{n}}}_{n = 1}^{\infty}$

Let $x = (1 + 2 n)^{\frac{1}{n}}$ .

\begin{aligned} lim_{n \to \infty} (1 + 2 n)^{\frac{1}{n}} & = x \\ ⟹ lim_{n \to \infty} \ln (1 + 2 n)^{\frac{1}{n}} & = \ln x \\ lim_{n \to \infty} \frac{1}{n} \ln (1 + 2 n) & = \ln x \\ ⟹ lim_{n \to \infty} \frac{1}{1 + 2 n} \cdot 2 & = \ln x \\ 0 & = \ln x \\ x & = 1 \end{aligned}

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