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Section 1.2

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1. (a) \displaystyle{a_n = \frac{1}{3^n}, n=0,1,2,\ldots}

(b) \displaystyle{a_n = \frac{1}{n}, n=1,2,3, \ldots}

(c) \displaystyle{a_n = \frac{2n+1}{n+1}, n=0,1,2,\ldots}

(d) \displaystyle{a_n = \frac{(-1)^n}{2n+1}, n=1,2,3,\ldots}

2. (a) Converges: \displaystyle{\lim_{n \to \infty}\frac{1}{3^n} = 0}

(b) Diverges: \displaystyle{\lim_{n \to \infty}\pi^n = \infty}

(c) Converges: \displaystyle{\lim_{n \to \infty}\frac{3n-1}{2n+6} = \frac{3}{2}}

(d) Diverges

(e) Diverges: \displaystyle{\lim_{n \to \infty}\frac{3n^4-6n^3+1}{5n^3+n^2+2}=\infty}

(f) Converges: \displaystyle{\lim_{n \to \infty}\frac{2n^5-3n^2+23}{7n^5+13n^4-12} = \frac{2}{7}}

(g) Converges: \displaystyle{\lim_{n \to \infty}\frac{45-16n^2}{13+5n+6n^3} = 0}

(h) Converges: \displaystyle{\lim_{n \to \infty}\frac{3n+1}{\sqrt{4n^2+1}}=\frac{3}{2}}

(i) Diverges: \displaystyle{\lim_{n \to \infty}(-2)^{2n+1} = -\infty}

(j) Divergs: \displaystyle{\lim_{n \to \infty}\frac{10-16n^3}{1+n^2}=-\infty}

(k) Converges: \displaystyle{\sqrt{\frac{3n^2+n-6}{5n^2+16}}=\sqrt{\frac{3}{5}}}

(l) Converges: \displaystyle{\lim_{n \to \infty}\frac{(-1)^n}{5^n}=0}

3. \displaystyle{\lim_{n \to \infty}\frac{\sin(n)}{n}=0}

4. (a) a_1 = 2 , a_2 = 2.25 , a_3 = 2.3704 , a_4 = 2.4414 , and a_5 = 2.4883 , where the last three have been rounded to four decimal places.

(e) a_{74} > 2.7 , but a_n < 2.7 for n < 74 .

(f) n = 135

5. (e) n = 13

(f) n = 41

7. (c) \displaystyle{a_9 = a_{10} = \frac{1562500}{567}}

11. (a) \displaystyle{N = \left\lfloor \frac{1}{\epsilon}\right\rfloor} ; for \epsilon = 0.001 , N = 1000

(b) \displaystyle{N = \left\lfloor \frac{\log(\epsilon)}{\log(0.98)}\right\rfloor} ; for \epsilon = 0.001 , N = 341

(c) \displaystyle{N = \left\lfloor \frac{1}{\sqrt{\epsilon}}\right\rfloor} ; for \epsilon = 0.001 , N = 31

(d) \displaystyle{N = \left\lfloor \frac{1}{\sqrt[3]{\epsilon}}\right\rfloor} ; for \epsilon = 0.001 , N = 10

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