AM1-Tema-03

1. Folosind definiţia limitei unui şir sau criteriul de convergenţă cu \varepsilon să se arate că

  1.     \[\lim\limits_{n\to\infty} \frac{2n}{n+1} = 2\]

  2.     \[\lim\limits_{n\to\infty} \frac{1}{2^n} = 0\]

2. Calculaţi limita şirului cu termenul general a_n, dacă

  1.     \[a_n = \sqrt{2n^2 + n - 1} - n, n\ge 1\]

  2.     \[a_n = \sqrt{n^2 + 2n} - n, n\ge 1\]

  3.     \[a_n = \sqrt{2n+1} - \sqrt{n+2}, n\ge 1\]

  4.     \[a_n = \sqrt{3n+1} - 2\sqrt{2n+1} + \sqrt{n+1}, n\ge 1\]

  5.     \[a_n = \frac{2^{n+1}}{\left( 2+\frac{1}{n} \right)^n}, n\ge 1\]

  6.     \[a_n = \sum\limits_{k=1}^n \frac{k^2}{n^3 + k}, n\ge 1\]

  7.     \[a_n = \sum\limits_{k=1}^n \frac{C_n^k}{k+2^n}, n\ge 1\]

  8.     \[a_n = \sum\limits_{k=1}^n \frac{1}{\sqrt{n^2 + k}}, n\ge 1\]

  9.     \[a_n = \sum\limits_{k=1}^n \frac{k}{n^2 + k}, n\ge 1\]

  10.     \[a_n = \left(\frac{\alpha n+1}{\alpha n - 1}\right)^n, n\ge 1 \text{ unde }\alpha \in \mathbb{R}.\]

3. Fie a \in \left(0,\infty\right). Să se calculeze \lim\limits_{n\to\infty} x_n, unde

    \[x_n = \frac{2^n}{2^{n+1} + a^n}, {n\ge 1}.\]