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

Question

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

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Answer 1 · 2018-06-14T15:30:56

The answer is $C$

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!