Does $int_1^infty f(x)ln(x)dx$ converge if $int_1^infty f(x)dx $ converges?
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Suppose a function $f:[1,infty)tomathbb R$ is such that $int_1^infty f(x),dx $ converges. Is it possible that $$int_1^infty f(x)ln(x),dx $$ diverges? I have a hard time finding such a function.
Edit: no idea why, but I had just thought naively (without checking) that $int frac 1{xln^k(x)},dx $ diverges for all $k$ just because $int frac 1{xln(x)},dx $ diverges. Sorry!
calculus integration convergence
asked Nov 24 at 16:16
Wolfgang
320612
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up vote
-3
down vote
favorite
Suppose a function $f:[1,infty)tomathbb R$ is such that $int_1^infty f(x),dx $ converges. Is it possible that $$int_1^infty f(x)ln(x),dx $$ diverges? I have a hard time finding such a function.
Edit: no idea why, but I had just thought naively (without checking) that $int frac 1{xln^k(x)},dx $ diverges for all $k$ just because $int frac 1{xln(x)},dx $ diverges. Sorry!
calculus integration convergence
asked Nov 24 at 16:16
Wolfgang
320612
1
Below you will find a hint. Please show your effort and share with us your thoughts.
– Robert Z
Nov 24 at 16:21
add a comment |
up vote
-3
down vote
favorite
up vote
-3
down vote
favorite
Suppose a function $f:[1,infty)tomathbb R$ is such that $int_1^infty f(x),dx $ converges. Is it possible that $$int_1^infty f(x)ln(x),dx $$ diverges? I have a hard time finding such a function.
Edit: no idea why, but I had just thought naively (without checking) that $int frac 1{xln^k(x)},dx $ diverges for all $k$ just because $int frac 1{xln(x)},dx $ diverges. Sorry!
calculus integration convergence
asked Nov 24 at 16:16
Wolfgang
320612
Suppose a function $f:[1,infty)tomathbb R$ is such that $int_1^infty f(x),dx $ converges. Is it possible that $$int_1^infty f(x)ln(x),dx $$ diverges? I have a hard time finding such a function.
Edit: no idea why, but I had just thought naively (without checking) that $int frac 1{xln^k(x)},dx $ diverges for all $k$ just because $int frac 1{xln(x)},dx $ diverges. Sorry!
calculus integration convergence
calculus integration convergence
asked Nov 24 at 16:16
Wolfgang
320612
asked Nov 24 at 16:16
Wolfgang
320612
edited Nov 24 at 19:54
asked Nov 24 at 16:16
Wolfgang
320612
asked Nov 24 at 16:16
Wolfgang
320612
asked Nov 24 at 16:16
Wolfgang
320612
320612
1
Below you will find a hint. Please show your effort and share with us your thoughts.
– Robert Z
Nov 24 at 16:21
add a comment |
1
Below you will find a hint. Please show your effort and share with us your thoughts.
– Robert Z
Nov 24 at 16:21
1
1
Below you will find a hint. Please show your effort and share with us your thoughts.
– Robert Z
Nov 24 at 16:21
Below you will find a hint. Please show your effort and share with us your thoughts.
– Robert Z
Nov 24 at 16:21
add a comment |
1 Answer
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Hint. Consider the function $$f(x)=frac{1}{xln^2(1+x)}.$$
Is $int_1^infty f(x)dx$ convergent? What about $int_1^infty f(x)ln(x) dx$?
answered Nov 24 at 16:19
Robert Z
92.7k1060130
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Hint. Consider the function $$f(x)=frac{1}{xln^2(1+x)}.$$
Is $int_1^infty f(x)dx$ convergent? What about $int_1^infty f(x)ln(x) dx$?
answered Nov 24 at 16:19
Robert Z
92.7k1060130
add a comment |
up vote
1
down vote
accepted
Hint. Consider the function $$f(x)=frac{1}{xln^2(1+x)}.$$
Is $int_1^infty f(x)dx$ convergent? What about $int_1^infty f(x)ln(x) dx$?
answered Nov 24 at 16:19
Robert Z
92.7k1060130
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Hint. Consider the function $$f(x)=frac{1}{xln^2(1+x)}.$$
Is $int_1^infty f(x)dx$ convergent? What about $int_1^infty f(x)ln(x) dx$?
answered Nov 24 at 16:19
Robert Z
92.7k1060130
Hint. Consider the function $$f(x)=frac{1}{xln^2(1+x)}.$$
Is $int_1^infty f(x)dx$ convergent? What about $int_1^infty f(x)ln(x) dx$?
answered Nov 24 at 16:19
Robert Z
92.7k1060130
answered Nov 24 at 16:19
Robert Z
92.7k1060130
answered Nov 24 at 16:19
Robert Z
92.7k1060130
answered Nov 24 at 16:19
Robert Z
92.7k1060130
92.7k1060130
add a comment |
add a comment |
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1
Below you will find a hint. Please show your effort and share with us your thoughts.
– Robert Z
Nov 24 at 16:21