study the convergence of v_n = Σ_(k=1) ^n (((−1)^(k+1) )/(2^k +ln(k)))


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Question Number 40123 by maxmathsup by imad last updated on 15/Jul/18

$s t u d y t h e c o n v e r g e n c e o f$ $v_{n} = \sum_{k = 1}^{n} \frac{{(- 1)}^{k + 1}}{2^{k} + l n (k)}$
	Answered by math khazana by abdo last updated on 19/Jul/18

	$w e h a v e v_{n} = \frac{1}{2} + \sum_{k = 2}^{n} \frac{{(- 1)}^{k + 1}}{2^{k} + l n (k)} \Rightarrow$ $∣ v_{n} ∣ ⩽ \frac{1}{2} + \sum_{k = 2}^{n} \frac{1}{2^{k} + l n (k)} b u t f o r k \in [[2, n]]$ $l n (k) ⩾ l n (2) > 0 \Rightarrow 2^{k} + l n (k) > 2^{k} \Rightarrow$ $\frac{1}{2^{k} + l n (k)} < \frac{1}{2^{k}} \Rightarrow∣ v_{n} ∣ < \frac{1}{2} + \sum_{k = 2}^{n} \frac{1}{2^{k}} = \sum_{k = 1}^{n} \frac{1}{2^{k}}$ $t h e s e r i e \sum_{k ⩾ 1} \frac{1}{2^{k}} i s c o n v e r g e n t s o (v_{n}) c o n v e r g e s$
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