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!