find-the-radius-of-n-1-ln-n-n-3-n-1-z-n-

Question Number 33354 by caravan msup abdo. last updated on 15/Apr/18

f i n d t h e r a d i u s o f \sum_{n ⩾ 1} \frac{l n (n)}{\sqrt{n^{3} + n + 1}} z^{n}

Commented by math khazana by abdo last updated on 20/Apr/18

t h i s s e r i e i s p o s i t i f s t e r m s l e t p u t u_{n} = \frac{l n (n)}{\sqrt{n^{3} + n + 1}}

u_{n} = \frac{l n (n)}{\sqrt{n^{3}} (\sqrt{1 + \frac{1}{n^{2}} + \frac{1}{n^{3}}})} \sim \frac{l n (n)}{n^{\frac{3}{2}} (1 + \frac{1}{2} (\frac{1}{n^{2}} + \frac{1}{n^{3}}))}

\sim \frac{l n (n)}{n^{\frac{3}{2}}} = v_{n} w e h a v e \frac{v_{n + 1}}{v_{n}} = \frac{\frac{l n (n + 1)}{{(n + 1)}^{\frac{3}{2}}}}{\frac{l n (n)}{n^{\frac{3}{2}}}}

= \frac{l n (n + 1)}{l n (n)} {(\frac{n}{n + 1})}^{\frac{3}{2}} \to 1 (n \to + \infty) s o t h e r a d i u s o f

c o n v e r g e n c e i s R = 1 .