Question

Use the integral test to determine whether the series converges or diverges:
∑[n = 2,∞,(1)/(n√(n^(2) - 1))]

Answer 1

\[\int_2^{\infty}\frac{1}{x\sqrt{x^2-1}}dx\] is what you want i guess

Answer 2

which luckly happens to be
\[-\tan^{-1}(x)\] and so it converges

Answer 3

Oh! I didn't catch that. Thank you!

Answer 4

yw
btw you can just about do this with your eyeballs because even though the denominator is not a polynomial, if you ignore the -1 under the radical it is pretty clear that the denominator behaves like n^2 and so you can use the comparison test. but if you told to use integral test this would do it

Answer 5

Comparison test as in the direct comparison test, right? Also, I was just checking my notes. Isn't the integral actually \[\sec ^{-1}n\] ?