Question

Please help! Use the integral test to determine if the series is convergent or divergent.

Answer 1

Convergent

Explanation:

∑₂°° 1 / (k (ln k)³)

Applying integral test:

∫₂°° 1 / (x (ln x)³) dx

If u = ln x, then du = 1/x dx.

∫ (1 / u³) du

-1 / (2u²) + C

Substitute back:

-1 / (2 (ln x)²) + C

Evaluate between the limits:

-1 / (2 (ln ∞)²) − -1 / (2 (ln 2)²)

1 / (2 (ln 2)²)

The integral converges, therefore the series must also converge.