Section 5.3 « Difference Equations to Differential Equations

Section 5.3

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1. (a) The series converges because it is a multiple of a geometric series with ratio $r = \dfrac{1}{5}$ . Moreover, $\displaystyle{\sum_{n=0}^\infty \frac{3}{5^n} = \frac{15}{4}}$

(c) The series converges because it is the sum of two geometric series with ratios $r =\dfrac{1}{2}$ for the first and $r = \dfrac{1}{3}$ for the second. Moreover, $\displaystyle{\sum_{n=1}^\infty \left(\frac{5}{2^n} + \frac{2}{3^n}\right) = 6}$

(e) Since $\displaystyle{\lim_{n \to \infty}\left(1 - \frac{1}{n}\right) = 1}$ , the series diverges by the $n$ th term test for divergence.

(g) Since $\dfrac{(-1)^n}{1000}$ does not have a limit as $n \rightarrow \infty$ , the series diverges by the $n$ th term test for divergence.

2. (a) The series converges because it is a multiple of a $p$ -series with $p = 3$ .

(c) The series converges because it is the sum of multiples of two $p$ -series with $p = 2$ for the first and $p = 4$ for the second.

(e) The series diverges because it is a $p$ -series with $p = \dfrac{1}{3}$ .

(g) The series diverges because it is a multiple of a $p$ -series with $p = \dfrac{1}{2}.$

5. (a) The series converges.

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Difference Equations to Differential Equations

Section 5.3

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