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By Douady R., Douady A.

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6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. ∗ 18. 1 n n 1 n n 1. Prove that x > y implies ln x > ln y. Is ln x bounded? 1 Prove that n+1 < ln(1 + 1/n) < n1 . x Prove that 1+x < ln(1 + x) < x. 2 (Fermat) for a = 1/2, 1/3, 2/3. Prove the unboundedness of lnnn . n 1 Compare 1 + n1 and (1 + n+1 )n+1 . n Prove the monotonicity of ln n . n−1 n−1 Prove that k=2 k1 < ln n < k=1 k1 .

N ln n − n + 1 2 ln n + (1 − D∞ ) + n 1 d(x ln x − x) = θ 2n . Now we know that 1 ≥ (1 − D∞ ) ≥ 21 , but it is possible to evaluate the value of √ D∞ with more accuracy. Later we will prove that 1 − D∞ = 2π. 4 asymptotics of sums 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. Problems. ∞ Does k=1 sin k converge? ∞ Does k=1 sin k 2 converge? 2 1 1 + 3n+1 + 3n+2 − .... Evaluate 1 + 21 − 23 + 14 + 15 − 26 + · · · − 3n ∞ an+1 Prove: If lim an < 1, then k=1 ak converge. ∞ Prove: If k=1 |ak − ak−1 | < ∞, then√{ak } converges.

14. 15. Problems. ∞ Does k=1 sin k converge? ∞ Does k=1 sin k 2 converge? 2 1 1 + 3n+1 + 3n+2 − .... Evaluate 1 + 21 − 23 + 14 + 15 − 26 + · · · − 3n ∞ an+1 Prove: If lim an < 1, then k=1 ak converge. ∞ Prove: If k=1 |ak − ak−1 | < ∞, then√{ak } converges. [ k] ∞ (−1) Prove the convergence of k=1 . k ∞ 1 Prove the convergence of k=2 ln3 k . ∞ Prove the convergence of k=2 k ln k√1ln ln k . ∞ 1 Prove the convergence of k=2 k ln k(ln ln k)2 . ∞ 1 Prove the convergence of k=2 k ln k and find its asymptotic formula.

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