Convergence of Sequences Worksheet with Answer Key

What's on the worksheet

1. Find the limit of each sequence \a_n\, or explain how it diverges.
   a_n = \dfrac{6 - n^3}{2n^2 + 1}a_n = \dfrac{\sqrt{9n^2 + 2}}{4n - 1}a_n = \dfrac{(-2)^n + 3^{n+1}}{3^n - 1}a_n = \dfrac{(-3)^n}{2^n + 1}
2. Find each limit by passing to a function of a real variable x, or by continuity.
   \lim_{n \to \infty} n \ln\left(1 + \dfrac{3}{n}\right)\lim_{n \to \infty} n^2\left(1 - \cos\dfrac{1}{n}\right)\lim_{n \to \infty} \cos\left(\dfrac{\pi n}{4n + 1}\right)\lim_{n \to \infty} \big(\ln(n^2 + 1) - \ln n\big)
3. Find the limit.
4. Use the squeeze theorem.
5. Let a_n = 3√(n) + 3√(n).
6. Decide whether each statement is true or false.
7. Let a_1 = 1 and a_n+1 = 65 - a_n.