Studiați natura seriilor:

1. \sum\limits_{n=1}^{\infty}\frac{n^2-2n+2}{\sqrt{n^6+6n^3+3}};

2. \sum\limits_{n=1}^{\infty}\frac{n\sqrt[3]{n}+1}{n^2+4n+3};

3. \sum\limits_{n=1}^{\infty}\frac{1}{\sqrt{n}}\left(\frac{2}{7}\right)^n;

4. \sum\limits_{n=1}^{\infty}(n^2+1)\ln\left(1+\frac{1}{n^2+1}\right);

5. \sum\limits_{n=1}^{\infty}\frac{5^n}{n^3+5^n};

6. \sum\limits_{n=1}^{\infty}\frac{1}{2^n+n};

7. \sum\limits_{n=0}^{\infty}\textrm{arctg }\frac{1}{n^2+3n+3};

8. \sum\limits_{n=1}^{\infty}\sqrt{n}e^{-(n^2+n)};

9. \sum\limits_{n=1}^{\infty}\ln\frac{n^2+3}{n^2+2};

10. \sum\limits_{n=1}^{\infty}2^n\sin\frac{1}{3^n};

11. \sum\limits_{n=1}^{\infty}\frac{(n+1)^{n-1}}{n^{n+1}};

12. \sum\limits_{n=2}^{\infty}\frac{1}{\ln n};

13. \sum\limits_{n=2}^{\infty}\left(\sqrt[n]{n}-1\right)^n;

14. \sum\limits_{n=1}^{\infty}\frac{n}{\left(1+\frac{1}{n}\right)^{n^2}};

15. \sum\limits_{n=1}^{\infty}e^{-\frac{n}{3}}\left(\frac{n+1}{n}\right)^{n^3};

16. \sum\limits_{n=1}^{\infty} n^2\sin\frac{\pi}{2^n};

17. \sum\limits_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!}\frac{1}{2n+1};

18. \sum\limits_{n=0}^{\infty}\frac{n!}{2\dot 5\dot 8\dot ...\dot (3n+2)};

19. \sum\limits_{n=1}^{\infty}7^{\ln n};

20. \sum\limits_{n=1}^{\infty}\frac{\cos{2n}}{n^2};

21. \sum\limits_{n=1}^{\infty}\frac{n\ln n}{(n+1)^3} (indicație: criteriul de comparație, apoi criteriul integral al lui Cauchy);

22. \sum\limits_{n=2}^{\infty}\frac{1}{n\ln n} (indicație: criteriul lui Cauchy de condensare);

23. \sum\limits_{n=1}^{\infty}(-1)^n\frac{(n+1)^{n+1}}{n^{n+2}} (indicație: criteriul lui Leibniz);

24. \sum\limits_{n=1}^{\infty}\frac{\sin n\cos n^2}{\sqrt{n}+\sqrt[3]{n}} (indicație: criteriul lui Dirichlet);

25. \sum\limits_{n=1}^{\infty}\frac{\sin n\cos\frac{1}{n}}{\sqrt{n+2}} (indicație: criteriile Dirichlet și, apoi, Abel).

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