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

Question

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

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

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