Question

Determine whether the improper integral converges or diverges, and find the value of each that converges.

∫^[infinity]_1 1/x^^0.999 dx

Answer 1

Divergent.

Explanation:

We have been given an integral
`\int\limits^\infty _1 {(1)/(x^(0.999)) \, dx`. We are asked to determine whether the given integral diverges or converges.

`\int _1^(\infty )\:\:\:(1)/(x^(0.999))\:\:\:dx`

`\int _1^(\infty )\:\:\:(1)/(x^(0.999))\:\:\:dx=\int _1^(\infty )\:\:\:x^(-0.999)\:\:\:dx`

`\int _1^(\infty )\:\:\:x^(-0.999)\:\:\:dx=(x^(-0.999+1))/(-0.999+1)`

`\int _1^(\infty )\:\:\:x^(-0.999)\:\:\:dx=(x^(0.001))/(0.001)`

`\int _1^(\infty )\:\:\:x^(-0.999)\:\:\:dx=1000x^(0.001)`

Let us compute the boundaries.

`1000(\infty)^(0.001)=\infty`

`1000(1)^(0.001)=1000`

Since
`\infty-1000` is not a finite number, therefore, the given integral diverges.