(Integral test). Let

f:{xR:x>=1}->R

be a decreasing positive-valued function. Prove that

\\\\sum_(n=1)^(\\\\infty ) f(n)

converges if and only if

\\\\lim_(n->\\\\infty )\\\\int_1^n f(x)dx

exists. (Hint: Draw a diagram.)\ Use the preceding problem to tell for which

\\\ ho >0

the following series converge:\

\\\\sum_(n=1)^(\\\\infty ) (1)/(n^(p)),\\\\sum_(n=1)^(\\\\infty ) (1)/(n(logn)^(p)),\\\\sum_(n=1)^(\\\\infty ) (1)/(nlogn(loglogn)^(p)).