Applying the Integral Test

Question

Use the Integral Test to determine if the series converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

∑(infinity) (n=1) (n/n^2 +4)

I graphed this as f(x) and the graph showed that it is positive and continuous at x≥1. However, the graph slightly increases before x=2 and then starts to decrease after x=2. Everything that I looked up shows that it does satisfy all conditions of the integral test and that I should proceed with solving. But how does it satisfy the conditions when it is not decreasing the entire time from x=1 onward?

Answer 1

y = x / (x²+4) increases on (1,2) and decreases on (2,∞). The function is positive and continuous for x > 0.

So, the Integral Test applies for n ≥ 2.

First, apply the integral test on the interval [2, ∞). If this series converges, then the given series also converges (adding a finite number of terms doesn't affect convergence). Similarly for divergence.

∫_{(2 to ∞)} [x / (x²+4)]dx = (1/2)lim_b→∞ [ln(b²+4) - ln8] = ∞

So, by the Integral Test, the series from n = 2 to ∞ diverges. Therefore, the series from n = 1 to ∞ also diverges.