(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)).

Integral test). Let f:{xR:x1}R be a decreasing positive-valued function. Prove that n=1f(n) converges if and only if limn1nf(x)dx exists.(Hint: Draw a diagram.) Use the preceding problem to tell for which >0 the following series converge: n=1np1,n=1n(logn)p1,n=1nlogn(loglogn)p1