find nature of the serie Σ (−1)^n U_n with U_(n+1) =(e^(−U_n ) /(n+1)) (U


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Question Number 90743 by abdomathmax last updated on 25/Apr/20

$f i n d n a t u r e o f t h e s e r i e Σ {(- 1)}^{n} U_{n}$ $w i t h U_{n + 1} = \frac{e^{- U_{n}}}{n + 1} (U_{0} = 1)$
	Answered by ~blr237~ last updated on 25/Apr/20

	$w e c a n s h o w b y i n d u c t i o n t h a t U_{n} \in] 0, 1]$ $s o Σ {(- 1)}^{n} U_{n} i s a n a l t e r n a t e d s e r i e t h e n i t c o n v e r g e s$ $i f (U_{n}) d e c r e a s e a n d c o n v e r g e s t o z e r o .$ $w e h a v e e^{- u_{n}} < u_{n}; \frac{1}{n + 1} ⩽ 1 t h e n u_{n + 1} < u_{n}$ $∣ u_{n} ∣⩽ \frac{1}{n} c a u s e e^{- u_{n}} < 1 . s o \lim_{n \to \infty} u_{n} = 0$
	Commented by mathmax by abdo last updated on 25/Apr/20

	$t h a n k x s i r$
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