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(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
(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
(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)).
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