688 Appendix C 11 Answers & Hints for Selected Exercises
(b) The contrapositive of (a).
(c) 2:(-l)n diverges, but (1 - 1) + (1-1) + (1 - 1) + · · · converges.
n n- Suppose An= L ak --t A and Bn = L kk --t B. Then, by the algebra of
k=l k=l
n n
limits of sequences, L (ak +bk) --t A+ Band L cak --t cA.
k=l k=l
EXERCISE SET 8.2- The sequence of partial sums of a nonnegative series is monotone increasing.
Apply Cor. 2.5.4. - Converges. Comparison test, with L ( ~) k.
- Diverges. Comparison test, with L 3 },,.
- Diverges. Comparison test, with L n~l.
- Converges. Ratio test; L = 1/2.
- Diverges. Root test; R = +oo.
- Diverges. Comparison test, with L ~-
- Converges by the integral test.
- Converges by comparison with Exercise 8.2.15.
- Diverges by the general term test.
- Converges. Ratio test; L = l/e.
- Diverges. Ratio test; L = +oo.
- Converges. Comparison test, with geometric series L (sin l l.
- Converges. Root test, R = 1/9.
- Diverges, since it has a regrouping that diverges: 1 + (Vs + ~) +
(Vs + -it4) + · · ·. The sequence of partial sums of this series is a subsequence
of the sequence of partial sums of L -ifk, a divergent p-series.- If p > 1, the integral test shows convergence. If p = 1, the integral test
shows divergence. If p < 1, use the comparison test with the p = 1 case. - (a) Suppose p > 1. Show that d~^1 ~:' < 0 if lnx > p; hence {^1 ~;'} is
eventually decreasing. By L'Hopital's rule,^1 ~:' --t 0, so the integral test applies,
to show that the series converges.