infinite series,comparision test,convergence divergence

enter image description here here as in example 3 the answer of the book is convergence but as (in round shape in photo ) i had solved and my answer is coming divergence by comparision test so what `s wrong in my answer (as divergence) so please can you clarify by doubt or misconcept here if any

• For one, your $u_n$ does NOT go to infinity. – Randall Oct 27 '17 at 13:16
• And, even if it did, how would this in tandem with the Comparison Test prove divergence? My gut tells me you are misunderstanding the Comparison Test. – Randall Oct 27 '17 at 13:17
• un=ln n/n now integration of (ln / n)dn limits 1 to infinity taking y=ln n so dy =1/n dn so now integration is ydy limits 0 to infinity which is y^2 / 2 limits 0 to infinity is so it will be infinity hence by integral test it diverges – Pritul Dave Oct 28 '17 at 0:16
• also vn = 1/sqrt(n) so by p test it is divergence so the whole series diverges – Pritul Dave Oct 28 '17 at 0:18

You have taken wrong Vn if $$Vn = n^{(\frac{-3}{2})}$$ than by p test it converge