91Ó°ÊÓ

State whether the sequence converges as \(n \rightarrow \infty\); if it does, find the limit. $$\frac{x^{100 n}}{n !}$$.

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

Determine the boundedness and monotonicity of the sequence with \(a_{n},\) as indicated. $$\frac{2^{n}-1}{2^{n}}$$.

Show by example (a) that the least upper bound of a set of rational numbers reed not be rational. (b) that the least upper bound of a set of irrational numbers need not be irrational.

(a) Show that the least upper bound of a set of negative numbers cannot be positive. (b) Show that the greatest lower bound of a set of positive numbers cannot be negative.

Determine the boundedness and monotonicity of the sequence with \(a_{n},\) as indicated. $$\cos n \pi$$.

Determine the boundedness and monotonicity of the sequence with \(a_{n},\) as indicated. $$\frac{(-2)^{n}}{n^{10}}$$.

Sketch the curves \(y=\sec x\) and \(y=\tan x\) for \(0 \leq x<\pi / 2\). Calculate the area between the two curves.

Show that linear combinations and products of bounded sequences are bounded. Sequences can be desired recursively: one or more terms are given explicitly; the remaining ones are then defined in terms of their predecessors.

Give the first six terms of the sequence and then give the \(n\) th term. $$a_{1}=1 ; \quad a_{n+1}=\frac{1}{2} a_{n}+1$$.