Using the exp-log technique,



We have ln(k) = ln(k/n) + ln(n), and so we can rewrite the limand as

Then the first sum converges to a definite integral,

while the remaining terms vanish since

So the limit is
.
Since

it's easy to show that the integral reduces to

so we can write the sum as

We have k²/(k² - 1) > 1 for all k, so that ln(k²/(k² - 1)) > ln(1) = 0. We can see the first 3 terms of the sum already exceed 1 > 49/50, so (D) is true.
Numerical computation of the sum suggests I < ln(99), but I have yet to come up with an analytical solution for this bound...