91Ó°ÊÓ

The first several terms of a sequence \(a_{1} \cdot a_{2} \cdots\) are given. Assume that the pattern continues as indicated and find an explicit formula for the \(a_{n}\). $$\text { 2, } \frac{5}{2}, \frac{10}{3}, \frac{17}{4}, \frac{26}{5}, \cdots$$.

Calculate. $$\lim _{x \rightarrow \pi / 2} \frac{\cos x}{\sin 2 x}$$

Find the least upper bound (if it exists) and the greatest lower bound (it if exists). $$\left\\{x: x^{2} < 4\right\\}$$.

Calculate. $$\lim _{x \rightarrow a} \frac{x-a}{x^{n}-a^{n}}$$

The first several terms of a sequence \(a_{1} \cdot a_{2} \cdots\) are given. Assume that the pattern continues as indicated and find an explicit formula for the \(a_{n}\). $$-\frac{1}{4}, \frac{2}{9},-\frac{3}{16}, \frac{4}{25},-\frac{5}{36}, \dots$$.

State whether the sequence converges as \(n \rightarrow \infty\); if it does, find the limit. $$\frac{3^{n}}{4}$$

Calculate. $$\lim _{x \rightarrow \infty} \frac{\ln x^{2}}{x}$$

State whatever the sequence converges and, if it does, find the limit. $$\frac{n+(-1)^{n}}{n}$$

Below we list some improper integrals. Determine whether the integral converges and, if so, evaluate the integral. $$\int_{0}^{1} \frac{d x}{\sqrt{x}}$$

The first several terms of a sequence \(a_{1} \cdot a_{2} \cdots\) are given. Assume that the pattern continues as indicated and find an explicit formula for the \(a_{n}\). $$\text { 1, } \frac{1}{2}, 3, \frac{1}{4}, 5, \frac{1}{6}, \cdots$$.