For all a 0 and b 1, the inequalities ln n n a , n a n are true for n sufficiently large (this can be proved using LHopitals Rule) Use this, together with the Direct Comparison Test, to determine whether the series converges or diverges 00 2 n 1 n n 1 3n...

Choose N such that n 2 for n N Then n 3n nN 3 nN ...

For all a > 0 and b > 1, the inequalities ln n n a ,

Question:

For all a > 0 and b > 1, the inequalities ln n ≤ n^a, n^a n are true for n sufficiently large (this can be proved using L’Hˆopital’s Rule). Use this, together with the Direct Comparison Test, to determine whether the series converges or diverges.

00 2: n=1 n n 1 3n