Question

Does this converges or diverges? integral 1 to infinity x^-1/3 dx

Answer 1

I got \[\frac{ 3 }{ 2 } \lim_{b \to infinity} (b ^{2/3} - 1^{2/3})\]

Answer 2

which = infinity. therefore it diverges

Answer 3

okay

Answer 4

\[\int\limits_{1}^{t}x ^{\frac{ -1 }{ 3 }}dx=\[\lim_{t \rightarrow \infty}\int\limits_{1}^{t} x ^{\frac{ -1 }{ 3 }}dx= \lim_{t \rightarrow \infty}\left[ \frac{ x ^{\frac{ 2 }{3 }} }{\frac{ 2 }{ 3 } } \right]_{1 }^{t}
\[=\frac{ 3 }{ 2 }\lim_{t \rightarrow \infty} \left( \left( t \right)^{\frac{ 2 }{ 3 }}-1 \right)\rightarrow \infty \] \]
therefore it diverges.