For which pairs of real numbers (p,q)(p, q)(p,q) does the Bertrand-like series ∑n=3∞1n(lnn)p(lnlnn)q\sum_{n=3}^{\infty} \frac{1}{n (\ln n)^p (\ln \ln n)^q}∑n=3∞n(lnn)p(lnlnn)q1 converge?
Converges if p>1p > 1p>1 for any qqq, or if p=1p = 1p=1 and q>1q > 1q>1.
Converges if p≥1p \ge 1p≥1 and q>1q > 1q>1.
Converges only if p>1p > 1p>1 and q>1q > 1q>1.
Converges if p+q>2p + q > 2p+q>2.