Section 5.4

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1. (a) The series converges because it is a multiple of a geometric series with ratio $r = \dfrac{1}{2}$ .

(e) The series converges because it is a geometric series with ratio $r = \dfrac{1}{\pi}$ .

(g) The series converges because is the sum of a convergent geometric series (with ratio $r = \dfrac{1}{3}$ ) and a convergent $p$ -series (with $p = \dfrac{3}{2}$ ).

2. (a) The series diverges by comparison with the harmonic series.

(e) The series converges by the limit comparison test with a geometric series with ratio $r = \dfrac{1}{3}$ .

(g) The series diverges by the limit comparison test with a $p$ -series with $p = 2$ .

4. The series $\displaystyle{\sum_{n=1}^\infty \frac{1}{4n^5 - 2}}$ converges by the limit comparison test with a $p$ -series with $p = 5$ . Hence $\displaystyle{\int_1^\infty \frac{1}{4x^5 - 2} \ dx}$ convergess by the integral test.

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Difference Equations to Differential Equations

Section 5.4

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