### Differential Equations to Difference Equations

Chapter 1

Section 1.1

4. (a) $y = 4(x - 1) + 2$

(b) $y = 2(x - 1) + 2$

(c) $y = 3(x - 1) + 1$

(d) $y = 4(x - 2) + 4$

5. For $6$ digits, $n = 5624$.

6. $n \ge 201$

7. (a) $p = 101$

(b) $|m_n - 2| < 0.01$ when $n > p$

8. (a) $a_{20} = 0.9996$ to four decimal places.

(c) $a_{20} = 2.6533$ to four decimal places.

(d) $a_{20} = 41.1032$ to four decimal places.

Section 1.2

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 $\epsilon = 0.001$, $N = 10$

Section 1.3

1. (a) $\displaystyle{\sum_{n=1}^\infty \left(\frac{1}{3}\right)^{n-1} = \frac{3}{2}}$

(c) $\displaystyle{\sum_{n=1}^\infty \left(\frac{2}{5^n}\right) = \frac{1}{2}}$

(e) $\displaystyle{\sum_{n=1}^\infty 7\left(\frac{1}{3}\right)^n\frac{2}{5}^{n-1} = \frac{35}{13}}$

(g) Does not converge

(i) Does not converge

(k) Does not converge

(m) $\displaystyle{\sum_{n=1}^\infty \sin(n\pi) = 0}$

(n) Does not converge

3. (a) $\pi$

(b) $99$

5. (a) $10$ billion dollars

(c) $\displaystyle{\frac{A}{1 - r}}$ dollars

7. $70$ meters; $\displaystyle{frac{50}{3}}$ meters

9. $\displaystyle{\frac{4}{9}}$

11. $83$ terms for a partial sum larger than $5$; $12,367$ terms for a partial sum larger than $10$

Section 1.4

1. (a) $x_0 = 10$, $x_1 = 14$, $x_2 = 18$, $x_3 = 22$, $x_4 = 26$, $x_5 = 30$

(c) $x_0 = 40$, $x_1 = 60$, $x_2 = 100$, $x_3 = 180$, $x_4 = 340$, $x_5 = 660$

(e) $x_0 = 2$, $x_1 = 3$, $x_2 = 5$, $x_3 = 8$, $x_4 = 13$, $x_5 = 21$ $x_6 = 34$

2. (a) $x_n = 5 \cdot 2^n$, $n = 0, 1, 2, \ldots$; $x_{10} = 5120$

(c) $x_n = 32.5(1.8)^n - 12.5$, $n = 0, 1, 2, \ldots$; $x_{10} = 11,591.5$, rounded to one decimal place

(e) $\displaystyle{x_n = \frac{10}{3} \cdot 4^n - \frac{4}{3}}$, $n = 0, 1, 2, \ldots$; $x_{10} = 3,495,252$

3. (a) $w_{n+1} = 1.03w_n$, $n = 0, 1, 2, \ldots$

(b) $w_n = 350(1.03)^n$, $n = 0, 1, 2, \ldots$ There will be $545$ weasels in $15$ years.

(d) $24$ years

(e) $\displaystyle{\lim_{n \to \infty}w_n = \infty}$ This says the population will grow without any upper bound to its size, which is not possible due to the physical limitations of the habitat.

5. (a) $w_n = 1.03w_n - 6$, $n = 0, 1, 2, \ldots$

(b) $w_n = 150(1.03)^n + 200$, $n = 0, 1, 2, \ldots$ There will be $434$ weasels in $15$ years.

(c) $\displaystyle{\lim_{n \to \infty}w_n = \infty}$ This says the population will grow without any upper bound to its size.

(d) The population will double in $41$ years.

7. (a) $155.30$ grams, rounded to two decimal places

(b) About $686$ years

(c) About $2,279$ years

9. (a) $\displaystyle{T_{n+1} = \frac{17}{19}T_n + \frac{140}{19}}$, $n = 0, 1, 2, \ldots$

(b) $\displaystyle{T_n = 95\left(\frac{17}{19}\right)^n + 70}$, $n = 0, 1, 2, \ldots$

(c) $75.9^\circ$ F, rounded to one decimal place

(d) $\displaystyle{\lim_{n \to \infty}T_n = 70^\circ}$ F

(f) No

11. $103.8^\circ$ F, rounded to one decimla place

15. $x_n = 4n + 10$, $n = 0, 1, 2, \ldots, 10$

Section 1.5

1. (a) $w_{n+1} = w_n + 0.00003w_n(1000 - w_n)$, $n = 0, 1, 2, \ldots$

(c) The population will double in $42$ years and triple in $102$ years.

(d) From the plot, it appears that $\displaystyle{\lim_{n \to \infty}w_n = 1000}$

3. (a) $p_{n+1} = 1.045p_n$, $n = 0, 1, 2, \ldots$

(c) The population will double in $16$ years and triple in $25$ years.

(d) $\displaystyle{\lim_{n \to \infty}p_n = \infty}$

(e) $52$ years

4. (a) It appears that $\displaystyle{\lim_{n \to \infty}r_n = 10,000}$

(c) There appears to be a limiting cycle of period $2$, with values alternating between $6,403$ and $11,930$.

(e) There appears to be a limiting cycle of period $8$, with values cycling thorough $4851$, $11245$, $7661$, $12248$, $5198$, $11588$, $6876$, and $12375$.

7. (a) $\displaystyle{\lim_{x \to \infty}x_n = \infty}$

(c) It appears that $\displaystyle{\lim_{n \to \infty}x_n = 0.7391}$ to four decimal places.

(e) It appears that $\displaystyle{\lim_{n \to \infty}x_n =\sqrt{2}}$.

(g) It appears that $\displaystyle{\lim_{n \to \infty}x_n = \infty}$.

(i) There appears to be a limiting cycle of order $2$.

(k) The limiting behavior, if any, is not clear.