rectangles ❯ ✮ ( ❘ ✑ ❯ ❚ ☞ ✑ ❯ ✒ ✕ ✽✩✶ ✌ ❯ ✄ ✑ ✮ and ✮ ✟ ✮ for each . Then ✑ ✙ ☎ ❱ ✮ . ➇ ➅ ✠ ✙ ✄ ❍➂✙✗✹➉ ✄ ✁ ✶ ❯ ✆✮ ✄✳✏✠➂✆✄✆✄✆✄ ✁ ❯ ) such that ✑ ❯ ❚ ❯✆ ☞✑ ❯ ✒ ✕ ✮ ✮ and ✮ ✟ and ✦ ➇ ✻➂❨➇ ➉ ❯ . Then 2. (a) Given a partition ➀ ❯ ✠ of✑ ➈ , let✑ ❯ ✄ and➇ ✗❯ ➂❨➇ be➉❁the inf and sup of ❵ on ➈ ❯ ✎ ② ✦ ➂ ➉ ❯ where ❯ ➈ ❯ ✎ ❯ ➈ ❯ ❯ , and the sum of the areas of the graph of ❵ is contained ✍✡✂ in ✑ these rectangles is ❷ ■❵ ■❵ . This can be made arbitrarily small (Lemma 4.5), so the

✟ ✡✥✞✜✒ ✏✓✽ ✟ ✟ ✟ ✡✖✏▼✡ ✛ ✟ ✒✛ ✟ ✡✥✙ ✏✓✽✠✙ ✱ ✱ 18. Diverges by Raabe’s test: ❚ 19. If ❚ ✱ ✆ ☛ ✱ ☛ ✱ ☛ ☛ ☛ ✣ ✽ ☛ ✽ ✰ ✱ ☛ ✱ ☛ ; series diverges by comparison ; series diverges by comparison to 16. ✏✓Converges ✒❨✽✠❂✮✕ ✄ ✴ by the extended root test (part (a) of Theorem 6.14): for large ✟ . 17. Converges by Raabe’s test: ✟ ✱ ✓✒ ☛ ✄ ☛ ✟ ✟ ❚ ✒ ✎ ☛ ✟ ➈ . ❃❝✡❋☞✎✍ ✏✓✒ ☛✾➉❊✽✠❂ q ✟ ✟ ✒ ☛❾☞♦❃❝✡ ✄ ☞✙ ✥ ✡ ✏✓✒❨✽✣☞ ✙ ✡ ☎✒ ✏☛✍⑩☞✳✱ ☛ ☛ ✮✽ ✱ ☛ ✒ ➉❴✄✕✞ ✽✣☞ ✙ ✡ ☎✒✰✣

the ✟ th partial sum is ✟ ❥ , so the full sum is 1. 7.3. Power Series 51 ✉❘ ✂ ✶✠✷✹✸ ✂❜✶ ✕✬✶✈➇ ✟ ❃ ➇ ✙ ✰ ✍ ✟ ❃ ➇ ✕ ✰ ✰ ➇❢✄✕✰ is if , if , and if . ☞❊➇❴✒✯✄ ✶✠✷✹✸ ✂❜✶ ✒ ✕ ✶✈➇✉✡✘✶✈➇❫✽✣☞✎✏❝✡✘✶ ✟ ➇ ✟ ✒ ➇t✄r ❶ ✰ ✏✓✽✩✶ ❘ (b) We have . If the second term tends to zero like ❵ ✔✟✞♣❘ ☞❊➇❫✒ ✟ ❃ ✰ ✶ ✣ ✥ ➇⑤✄❞✰ ✍ ✟❃ ➇✴✙⑦as✰ ❘ ✂ ❵✔ is➇✕✄ , ✰ , or it vanishes to begin with. Hence ✝ for , ➇✖✄ ✰ , and ➇ ✕ if ✰ ✒ 3. (a) Just observe that ✝ , or discontinuous. ✔✟✞ respectively. The convergence

✒✁✆ ✂☎☞❊✁✬✒■✍ ✂✈☞❊✢●✒✗ ✸✕ Given , if then . ✆ ✆ ✆ ✆ ✆ ❛ ✕ ❛ ✎ ❛ ☞✳✱✘✡✣✗✻✒ ✍ ✱ ✕ ✗ ☞✳✱✘✡❲❛❨✒ ✎ (a) ✝✟✞ . ✌ ✝✠✞ ✌ , so ➇❄➂❨➁ ➇✿ ✆ q✚☞❜ ➇⑧✍⑥➁❙ ❨✡⑤ ➁❙ ❺✒ ✆ q✚ ➇❧✍⑥➁❙ ✆ ✡ ➁❙ ✆ ➇ ➁ , and likewise with and switched; (b) For any ➇■ ✆ ✍t ➁❫we ✆ have q✚ ➇♣✍✇➁❙ ✆ hence ✡ ✡ . ✡ ✡ ✂☎☞❊✁✬✒☛✍ ✂☎☞❊✢●✒✗ ✕ ✁✿➂❨✢⑤✐ ✁◆✍⑩✢❤ ✕ ✙✕✰ ✱ ➂ ✙✌✰ ✟ ❷ and ✒ Given , we so ☛●☞❊✁✬✒✦ ✍ ☛■can ☞ ☞❊✢●✒✗choose ✕ ✟ ✵ ✵ ✹ ✁✿➂❨that ✢ ✐ ✁✥✍✖✢✂ ✟ ✕ whenever ✄ ✞ ✔ ✕❙☞ ✱➂ ✟ ✵ ❷ and ✵

interval about 0 contains small subintervals on which ✠ ✕ ✰ . ☎ ☞❊➇❴✒✗ ❬q✥➇ ✟ ➇ ☎ ☞♦✰✾✒☛✄ ✔✟✞ ☎✄✂ ☎ ☞❊➇❫✒❨✽✩➇⑥✄✖✰ since for all . ✝ ☞ ✱⑧✡ ☎ ✒✂✍ ✳☞ ✱✫✒ ➉❊✽ ☎ ✄ ✝☞ ✆✱✒ ✳ ✆★✐r✳☞ ✱❍➂ ✱❧✡ ☎ ✒ ☎ ✙❖✰ ☎ ✣ ✰ ✆ ✣ ✰ ❵ ✁❵ for some For✝☞ ✆✱✰ . As , also, and so ✒ ✣✟✞ , ➈❺❵ . ❵ ✠ ✍ 6. These formulas ✔✟✞✡✠ ☞✳✱✣✒❨✽ are ✄ ☞ ☛✍✌❜☞✳✱✫by ✒ applying ☞✳✱✫✒ general ✄ ☞✳✱ ✡ result ☞✳✱✣✒ ✁☛ ☛✠ ☛ ✠ l’Hˆopital’s rule 2 or 3 times. ☛ ✠ The ☎ ☛✖obtained ☎ ✒❝✍ is that ✂ is the operator defined by ✝ ❵ ☛ ☛✠