C cheffy Junior Member Joined Jan 10, 2007 Messages 73 Mar 11, 2007 #1 This is the last one, I think/hope! \(\displaystyle \[ \int_1^\infty \frac{\sqrt[3]{x^2+2}} {\sqrt{x^3+3}} \,dx.\]\) Converge or diverge? I have no idea how to simplify this. =\
This is the last one, I think/hope! \(\displaystyle \[ \int_1^\infty \frac{\sqrt[3]{x^2+2}} {\sqrt{x^3+3}} \,dx.\]\) Converge or diverge? I have no idea how to simplify this. =\
tkhunny Moderator Staff member Joined Apr 12, 2005 Messages 11,325 Mar 11, 2007 #2 Is it any easier like this? \(\displaystyle \sqrt[6]{\frac{(x^{2}+2)^{2}}{{(x^{3}+3)^{3}}}\)?
tkhunny Moderator Staff member Joined Apr 12, 2005 Messages 11,325 Mar 11, 2007 #4 Okay, then first guess from the way it is written... \(\displaystyle \frac{x^{2/3}}{x^{3/2}}\;=\;x^{-5/6}\) Since -1 < -5/6 < 1, I suspect it diverges. Now prove it.
Okay, then first guess from the way it is written... \(\displaystyle \frac{x^{2/3}}{x^{3/2}}\;=\;x^{-5/6}\) Since -1 < -5/6 < 1, I suspect it diverges. Now prove it.
C cheffy Junior Member Joined Jan 10, 2007 Messages 73 Mar 11, 2007 #5 Would I multiply out (x+2)^2 and then take the largest power and do the same as the bottom? And then compare that to the original?
Would I multiply out (x+2)^2 and then take the largest power and do the same as the bottom? And then compare that to the original?
