nature of the sequence u_n = Σ_(k=2) ^n (((−1)^k )/(kln(k))) .


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Question Number 29500 by abdo imad last updated on 09/Feb/18

$n a t u r e o f t h e s e q u e n c e u_{n} = \sum_{k = 2}^{n} \frac{{(- 1)}^{k}}{k l n (k)} .$
Commented by abdo imad last updated on 11/Feb/18

$l e t p u t v_{k} = \frac{1}{k l n (k)} f o r k ⩾ 2 w e h a v e {l i m}_{k \to \infty} v_{k} = 0 l e t$ $p r o v e t h a t {(v_{k})}_{k} i s d e c r e a s i n g w e p u t f (t) = \frac{1}{t l n (t)} f o r t ⩾ 2$ $f^{^{'}} (t) = - \frac{l n t + 1}{{(t l n (t))}^{2}} < 0 f o r x ⩾ 2 s o f i s d e c r e a z i n g a n d$ ${l i m}_{n \to + \infty} u_{n} = \sum_{k = 2}^{\infty} {(- 1)}^{k} v_{k} w i c h i s a a l t e r n a t e s e r i e$ $c o n v e r g e n t e s o (u_{n}) i s c o n v e r g e n t .$
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