Copyright\begin{matrix}c \\ ◯ \end{matrix}2004 ONG TCV Scoala Virtuala a Tanarului Matematician http://math.ournet.md 1

Limite de siruri

1. a) Utilizand definitia limitei cu ”ε” sa se arate ca sirul numeric a_{n} este convergent si are limita a.

b) Sa se determine rangul, incepand de la care termenul sirului difera de a cu mai putin de 0.001.

\begin{matrix}1.1. & a_{n} = \frac{4n + 2}{5n + 1}, & a = \frac{4}{5}· & 1.2. & a_{n} = \frac{2n + 1}{3n − 2}, & a = \frac{2}{3}· \\ 1.3. & a_{n} = \frac{1 − 2n}{n + 2}, & a = −2. & 1.4. & a_{n} = \frac{3n}{2n − 1}, & a = \frac{3}{2}· \\ 1.5. & a_{n} = \frac{7n + 1}{1 − 3n}, & a = −\frac{7}{3}· & 1.6. & a_{n} = \frac{2n − 1}{7n − 3}, & a = \frac{2}{7}· \\ 1.7. & a_{n} = \frac{2 − n}{1 − 2n}, & a = \frac{1}{2}· & 1.8. & a_{n} = \frac{6 − 3n}{2n − 1}, & a = −\frac{3}{2}· \\ 1.9. & a_{n} = \frac{2n^{2} + 1}{8n^{2} − 1}, & a = \frac{1}{4}· & 1.10. & a_{n} = \frac{2n^{2}}{3 − n^{2}} , & a = −2. \\ 1.11. & a_{n} = \frac{3n^{2} + 2}{1 − 4n^{2}}, & a = −\frac{3}{4}· & 1.12. & a_{n} = \frac{1 − 5n^{2}}{2 − 4n^{2}} , & a = \frac{5}{4}· \\ 1.13. & a_{n} = \frac{4n^{3}}{2n^{3} − 1}, & a = 2. & 1.14. & a_{n} = \frac{8 − n^{3}}{1 + 2n^{3}} , & a = −\frac{1}{2}· \\ 1.15. & a_{n} = \frac{2 + n^{3}}{2n^{3} − 1}, & a = \frac{1}{2}· & 1.16. & a_{n} = \frac{1 − n^{3}}{1 + n^{3}} , & a = −1. \end{matrix}

2. a) Sa se arate ca sirurile date sunt convergente:

2.1. a_{n} = \frac{2n + 3}{3n − 2}·  2.2. a_{n} = \frac{n − 1}{n + 1}·
2.3. a_{n} = \frac{2n + 1}{n + 3}·  2.4. a_{n} = \frac{1 − 2n}{n + 1}·
2.5. a_{n} = \frac{1 − 3n}{1 − 4n}·  2.6. a_{n} = \frac{2n − 1}{n + 2}·
2.7. a_{n} = ∑_{k=1}^{n} \frac{1}{k(k + 1)}·  2.8. a_{n} = ∑_{k=2}^{n} \frac{1}{k(k − 1)}·

2.9. a_{n} = ∑_{k=1}^{n} \frac{2}{(2k − 1)(2k + 1)}·  2.10. a_{n} = ∑_{k=1}^{n} \frac{1}{k^{2}} ·

2.11. a_{n} = \underbrace{\sqrt{2 + \sqrt{2 + … + \sqrt{2}}}}_{n rădăcini} .  2.12. a_{n} = \underbrace{\sqrt[3]{6 + \sqrt[3]{6 + … + \sqrt[3]{6}}}}_{n rădăcini} .

2.13. a_{n} = ∑_{k=1}^{n} \frac{sin k}{3^{k}} ·  2.14. a_{n} = ∑_{k=1}^{n} \frac{sin k!}{k(k + 1)}·

2.15. a_{n} = 1 + \frac{1}{2!} + ⋯ + \frac{1}{n!}·  2.16. a_{n} = \frac{1}{1 · 2} − \frac{1}{2 · 3} + ⋯ + \frac{(−1)^{n−1}}{n(n + 1)}·

Copyright\begin{matrix}c \\ ◯ \end{matrix}2004 ONG TCV Scoala Virtuala a Tanarului Matematician http://math.ournet.md 2

b) Sa se arate ca:

2.17. lim_{n→∞} sin \frac{π}{2} n ≠ 1.  2.18. lim_{n→∞} cos πn ≠ 1.
2.19. lim_{n→∞} \frac{2n − 1}{n + 1} ≠ 1.  2.20. lim_{n→∞} \frac{3n − 1}{2n + 1} ≠ 2.
2.21. lim_{n→∞} n^{2} sin \frac{πn}{4} ≠ 0.  2.22. lim_{n→∞} \frac{n}{n + 1} cos \frac{2πn}{3} ≠ 1.

3. Sa se calculeze urmatoarele limite:

3.1. lim_{n→∞} \frac{(n + 2)^{2} + (n − 1)^{2}}{(2n − 1)^{2} + (n + 1)^{2}} .  3.2. lim_{n→∞} \frac{(n + 1)^{2} − (n − 4)^{2}}{(3n + 1)^{2} + (n − 1)^{2}} .
3.3. lim_{n→∞} \frac{(2 − n)^{2} − (1 + n)^{2}}{(n − 3)^{2} − (n + 2)^{2}} .  3.4. lim_{n→∞} \frac{(2n − 1)^{2} − (n − 1)^{2}}{(n + 1)^{2} + (n − 1)^{2}} .
3.5. lim_{n→∞} \frac{(1 + 2n)^{3} − 8n^{3}}{(1 − 3n)^{2} − 3n^{2}} .  3.6. lim_{n→∞} \frac{(n + 3)^{3} + (n − 1)^{3}}{2n^{3} + 3n} .
3.7. lim_{n→∞} \frac{(n + 5)^{2} + (n + 2)^{2}}{(n + 2)^{3} − (n + 1)^{3}} .  3.8. lim_{n→∞} \frac{(2n + 1)^{2} + (1 − 3n)^{2}}{(n − 2)^{3} − (n − 1)^{3}} .
3.9. lim_{n→∞} \frac{(n + 2)^{4} − (n − 2)^{4}}{(n + 3)^{2} + (n − 3)^{2}} .  3.10. lim_{n→∞} \frac{(n + 1)^{4} − (n − 1)^{4}}{(n + 1)^{3} + (n − 1)^{3}} .
3.11. lim_{n→∞} \frac{(2n + 1)! + (2n + 2)!}{(2n + 3)!}.  3.12. lim_{n→∞} \frac{n! + (n + 2)!}{(n − 1)! + (n + 2)!}.
3.13. lim_{n→∞} \frac{(n + 3)! − (n + 1)!}{(n + 2)!}.  3.14. lim_{n→∞} \frac{(2n − 1)! + (2n + 1)!}{(2n)!(n + 1)}.
3.15. lim_{n→∞} \frac{(3n)! + (3n − 2)!}{(3n − 1)!(2n + 1)}.  3.16. lim_{n→∞} \frac{(n − 1)! + (n − 2)!}{(n − 3)!(3n^{2} − 1)}.

4. Sa se calculeze limitele:

4.1. lim_{n→∞} \sqrt{n + 1} (\sqrt{n + 3} − \sqrt{n + 2}) . 4.2. lim_{n→∞} (\sqrt{(n − 1)(n + 4)} − n) .
4.3. lim_{n→∞} (\sqrt{n^{2} + 3n + 2} − n) .  4.4. lim_{n→∞} (n + \sqrt[3]{n^{2} − n^{3}}) .
4.5. lim_{n→∞} (\sqrt{n^{2} + 4n − 2} − \sqrt{n^{2} − 2}) . 4.6. lim_{n→∞} \sqrt{n − 1} (\sqrt{n + 1} − \sqrt{n − 3}) .
4.7. lim_{n→∞} (n\sqrt{n} − \sqrt{n(n^{2} − 1)}) .  4.8. lim_{n→∞} n (\sqrt[3]{2 + 8n^{3}} − 2n) .
4.9. lim_{n→∞} \sqrt[3]{n} (\sqrt[3]{n^{2}} − \sqrt[3]{n(n + 1)}) .  4.10. lim_{n→∞} n^{2} (\sqrt[3]{n^{3} + 7} − \sqrt[3]{n^{3} + 1}) .
4.11. lim_{n→∞} (\frac{1}{n^{2}} + \frac{2}{n^{2}} + ⋯ + \frac{n − 1}{n^{2}}) .  4.12. lim_{n→∞} (\frac{2 + 4 + … + 2n}{n + 2} − n) .

4.13. lim_{n→∞} (\frac{n + 2}{1 + 2 + … + n} − \frac{3}{2}) .  4.14. lim_{n→∞} \frac{1 + 3 + 5 + … + 2n − 1}{2 + 4 + 6 + … + 2n}.

4.15. lim_{n→∞} \frac{5 + 10 + … + 5n}{n^{2} + 1}.  4.16. lim_{n→∞} \frac{1 · 2 + 2 · 3 + … + n(n + 1)}{n^{3}}.

4.17. lim_{n→∞} \frac{1^{2} + 3^{2} + … + (2n − 1)^{2}}{n^{3}} . 4.18. lim_{n→∞} (\frac{1^{2} + 2^{2} + … + n^{2}}{n^{2}} − \frac{n}{3}) .

4.19. lim_{n→∞} \frac{3^{n} − 5^{n+1}}{3^{n+1} + 5^{n+2}} .  4.20. lim_{n→∞} (\frac{7}{10} + \frac{29}{100} + ⋯ + \frac{2^{n} + 5^{n}}{10^{n}} ) .

4.21. lim_{n→∞} \frac{4^{n} + 7^{n}}{4^{n} − 7^{n−1}} .  4.22. lim_{n→∞} (\frac{3}{4} + \frac{5}{16} + ⋯ + \frac{1 + 2^{n}}{4^{n}} ) .

4.23. lim_{n→∞} \frac{3^{n} + 5^{−n}}{3^{−n} + 5^{n}} .  4.24. lim_{n→∞} \frac{1 + \frac{1}{5} + ⋯ + \frac{1}{5^{n}}}{1 + \frac{1}{7} + ⋯ + \frac{1}{7^{n}}} .

5. Sa se calculeze limitele:

5.1. lim_{n→∞} (\frac{2n + 3}{2n − 1})^{n}.  5.2. lim_{n→∞} (\frac{n + 2}{n + 1})^{1−n}.

5.3. lim_{n→∞} (\frac{3n − 1}{3n + 2})^{2n+1}.  5.4. lim_{n→∞} (\frac{2n + 1}{2n − 5})^{\frac{n}{6}} .

5.5. lim_{n→∞} (\frac{n^{2} − 1}{n^{2} + 1})^{2n−1}.  5.6. lim_{n→∞} (\frac{3n^{2} + 2}{3n^{2} − 1})^{\frac{n}{3}} .

5.7. lim_{n→∞} (\frac{3n + 1}{3n})^{1−n^{2}}.  5.8. lim_{n→∞} (\frac{2n^{2} + 1}{2n^{2} − 3})^{1−n^{3}}.

5.9. lim_{n→∞} (\frac{2n^{2} + 2}{2n^{2} + 1})^{n^{2}}.  5.10. lim_{n→∞} (\frac{n^{2} + n + 1}{n^{2} + n − 1})^{n^{2}−1}.

5.11. lim_{n→∞} (\frac{3n^{2} + 2}{3n^{2} − 1})^{n^{2}+1}.  5.12. lim_{n→∞} (\frac{2n^{2} + n + 2}{2n^{2} − 2n + 3})^{2n}.

5.13. lim_{n→∞} (\frac{3n^{2} + 6n + 7}{3n^{2} + 6n + 4})^{6n^{2}−5n+4}. 5.14. lim_{n→∞} (\frac{n^{2} + 1}{n^{2} − 1})^{2n^{2}}.

5.15. lim_{n→∞} (\frac{n^{2} + 2n + 3}{n^{2} + 3n + 4})^{2n−1}.  5.16. lim_{n→∞} (\frac{4n^{2} + 2}{4n^{2} − 2})^{n^{2}}.

5.17. lim_{n→∞} (\frac{n^{2} + n − 1}{3n^{2} − n + 1})^{\frac{1}{n}} .  5.18. lim_{n→∞} (\frac{1 + 2n}{3n − 1})^{n}.

5.19. lim_{n→∞} (\frac{n + 1}{n − 1})^{\frac{2n^{2}+1}{2n^{2}−1}} .  5.20. lim_{n→∞} (\frac{n^{2} − 1}{2n^{2} + 1})^{\frac{2n}{n+1}} .