1. **Problem 1: Find the limit of the sequence** $$a_n = \frac{(3n + 1)!}{(3n - 1)!}$$ The factorial expression can be expanded: $$\frac{(3n + 1)!}{(3n - 1)!} = (3n)(3n + 1)$$ because $$(3n + 1)! = (3n + 1)(3n)(3n - 1)!$$ So, $$a_n = (3n)(3n + 1) = 9n^2 + 3n$$ As $n \to \infty$, $9n^2 + 3n \to \infty$, so the sequence diverges. 2. **Problem 2: Determine convergence or divergence of sequences** (1) $$a_n = \frac{n^2 - 4}{n^2 + 3}$$ Divide numerator and denominator by $n^2$: $$a_n = \frac{1 - \frac{4}{n^2}}{1 + \frac{3}{n^2}}$$ As $n \to \infty$, terms with $\frac{1}{n^2} \to 0$, so $$\lim_{n \to \infty} a_n = \frac{1 - 0}{1 + 0} = 1$$ Sequence converges to 1. (2) $$a_n = \cos(7\pi n)$$ Since $\cos(7\pi n) = \cos(\pi n)$ because $7\pi n$ is an integer multiple of $\pi$, and $$\cos(\pi n) = (-1)^n$$ This sequence oscillates between 1 and -1, so it does not converge. Sequence diverges. (3) $$a_n = \cos\left(\frac{2}{n}\right)$$ As $n \to \infty$, $\frac{2}{n} \to 0$, so $$\lim_{n \to \infty} a_n = \cos(0) = 1$$ Sequence converges to 1. 3. **Problem 3: Limit of sequence** $$a_n = \frac{5n^2 + n + 2}{6n^2 - 3}$$ Divide numerator and denominator by $n^2$: $$a_n = \frac{5 + \frac{1}{n} + \frac{2}{n^2}}{6 - \frac{3}{n^2}}$$ As $n \to \infty$, terms with $\frac{1}{n}$ and $\frac{1}{n^2} \to 0$, so $$\lim_{n \to \infty} a_n = \frac{5 + 0 + 0}{6 - 0} = \frac{5}{6}$$ Sequence converges to $\frac{5}{6}$. 4. **Problem 1 (Geometric series):** Series: $$10 + 10\frac{e}{\pi} + 10\frac{e^2}{\pi^2} + 10\frac{e^3}{\pi^3} + \cdots$$ (a) General term: $$a_n = 10 \left(\frac{e}{\pi}\right)^n$$ with $n$ starting at 0. (b) Ratio: $$r = \frac{e}{\pi}$$ Since $e \approx 2.718$ and $\pi \approx 3.1415$, $r = \frac{e}{\pi} < 1$, so the series converges. (c) Sum of geometric series: $$S = \frac{a}{1 - r} = \frac{10}{1 - \frac{e}{\pi}} = \frac{10}{\frac{\pi - e}{\pi}} = \frac{10\pi}{\pi - e}$$ 5. **Problem 2 (Geometric series):** Series: $$5 + \frac{5}{6} + \frac{5}{36} + \frac{5}{216} + \cdots$$ (a) General term: $$a_n = 5 \left(\frac{1}{6}\right)^n$$ with $n$ starting at 0. (b) Ratio: $$r = \frac{1}{6}$$ Since $|r| < 1$, the series converges. (c) Sum: $$S = \frac{a}{1 - r} = \frac{5}{1 - \frac{1}{6}} = \frac{5}{\frac{5}{6}} = 6$$ 6. **Problem 3 (Geometric series):** Given $a_1 = 5$ and $$\frac{a_n}{a_{n+1}} = \frac{1}{4}$$ for $n \geq 1$. Rewrite ratio: $$\frac{a_n}{a_{n+1}} = \frac{1}{4} \implies a_{n+1} = 4 a_n$$ This means the sequence grows by factor 4 each step, so the terms are: $$a_1 = 5, a_2 = 20, a_3 = 80, \ldots$$ This is a geometric sequence with ratio $r = 4$. (a) Sum: $$\sum a_n = 5 + 20 + 80 + \cdots = \sum_{n=0}^\infty 5 \cdot 4^n$$ (b) Since $|r| = 4 > 1$, the series diverges. (c) Value: Diverges (no finite sum). **Final answers:** - Problem 1 sequence limit: Diverges - Problem 2 sequences: (1) Converges to 1, (2) Diverges, (3) Converges to 1 - Problem 3 sequence limit: $\frac{5}{6}$ - Geometric series 1 sum: $\frac{10\pi}{\pi - e}$ - Geometric series 2 sum: 6 - Geometric series 3 sum: Diverges