Chapter 8

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. Suppose that \(f\) is a continuous, positive, and decreasing function on $[1, \infty) .\( If \)f(n)=a_{n}\( for \)n \geq 1\( and \)\sum_{n=1}^{\infty} a_{n}$ is convergent, then $\sum_{n=1}^{\infty} a_{n} \leq a_{1}+\int_{1}^{\infty} f(x) d x$.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) and \(B\) are events of an experiment, then $$P(A \cap B)=P(A \mid B) \cdot P(B)=P(B \mid A) \cdot P(A)$$

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. \(\int_{1}^{\infty} \frac{d x}{x(x+1)}<\infty\).