Question

How do I determine if the improper integrals converge or not?

I narrowed the options down to A and C since Bk must converge for k > 1. How do I do the same for Ak?

Answer 1

The answer is `C`

Explanation:

Considering `A_k`

If the value of `k` is greater than `1`, you will have something of the form

`[1/x^k]_0^1`

This won't work, because `1/0` is undefined which makes the integral diverge.

This eliminates options A, D and E.

Considering `B_k`

This is your basic p-series. Just like the above integral yields something close to `[1/x^k]_0^1`, this one will be as follows:

`lim_(t->oo) [1/x^k]_1^t`

Recall that `lim_(t->oo) 1/t^k = 0` so this converges (has a finite value).

However, if we're of the form `0 < k < 1`, the numbers will be massive and the integral will diverge.

This means the correct answer is C.

Hopefully this helps!