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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

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  • $\begingroup$ For one, your $u_n$ does NOT go to infinity. $\endgroup$ – Randall Oct 27 '17 at 13:16
  • $\begingroup$ 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. $\endgroup$ – Randall Oct 27 '17 at 13:17
  • $\begingroup$ 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 $\endgroup$ – Pritul Dave Oct 28 '17 at 0:16
  • $\begingroup$ also vn = 1/sqrt(n) so by p test it is divergence so the whole series diverges $\endgroup$ – Pritul Dave Oct 28 '17 at 0:18
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You have taken wrong Vn if $$Vn = n^{(\frac{-3}{2})}$$ than by p test it converge

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