Section 5.6 « Difference Equations to Differential Equations

Section 5.6

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1. (a) The series converges absolutely ( $p$ -series with $p = 3$ ), and so converges.

$\displaystyle{\lim_{n \to \infty}\frac{3n^2 - 1}{4n^2 + 2} = \frac{3}{4}}$ .

(e) The series does not converge absolutely (limit comparison test with a $p$ -series with $p = \dfrac{1}{2}$ ), but does converge conditionally by Leibniz’s theorem. Hence the series converges.

(g) Since $\displaystyle{\lim_{n \to \infty}\frac{n!}{2^n} = \infty}$ , the terms $\dfrac{(-1)^n n!}{2^n}$ do not have a limit as $n \rightarrow \infty$ . Hence the series diverges by the $n$ th term test for divergence.

2. (a) The series converges absolutely (limit comparison with a $p$ -series with $p = 3$ ), and so converges.

(e) The series converges absolutely (ratio test, $\rho = 0$ ), and so converges.

(g) Since $\displaystyle{\lim_{n \to \infty}\frac{n + 1}{2n - 1} = \frac{1}{2}}$ , the terms $\dfrac{(-1)^n (n + 1)}{2n - 1}$ do not have a limit as $n \rightarrow \infty$ . Hence the series diverges by the the $n$ th term test for divergence.

3. (a) $\displaystyle{s_{15} = \frac{34,361,893,981}{93,405,312,000} = 0.367879441171397}$ , to 15 decimal places

(b) $|s - s_{15}| \le \dfrac{1}{16!} = 4.779 \times 10^{-14}$ , rounded to 17 decimal places

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Difference Equations to Differential Equations

Section 5.6

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