integral test question

Question

Use the integral test to determine if the series  is convergent or divergent. Make sure that you set up the problem correctly and use the proper notation.

Answer 1

f(x) = x/(x²+1)²

f(x) is continuous, positive and decreasing on the interval [1,∞)

If ∫ f(x) dx converges then ∑a_n converges.

∫₁^∞x/(x²+1)²dx = lim t→∞∫₁^tx/(x²+1)²dx = lim t→∞(1/2)(-1/(x²+1))|₁^t

= 1/2(-0 + 1/2) = 1/4

The integral converges to 1/4, therefore ∑a_n also converges