## Section 5.4

Please post comments if you find any errors, or if you wish to contribute answers for additional problems. You may find information on using $\LaTeX$ here.

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

(c) The series diverges by comparion with a $p$-series with $p = \dfrac{1}{2}$.

(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.

(c) The series converges by the limit comparison test with a $p$-series with $p = 3$.

(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.