Math Question and Answers


Question and Answers Forum

All Questions Topic List
None Questions
Previous in All Question Next in All Question
Previous in None Next in None
Question Number 154507 by RB95 last updated on 19/Sep/21

Commented by RB95 last updated on 19/Sep/21

$S a l u t$ $A i d e z - m o i s v p$ $u r g e n t$ $m e r c i$
	Answered by Jonathanwaweh last updated on 19/Sep/21

	$1) p_{n} = \underset{k = 1}{\prod^{n}} 2^{k^{2}} {l n p}_{n}) = \underset{k = 1}{\sum^{n}} l n (2^{k^{2}})$ $= \underset{k = 1}{\sum^{n}} k^{2} l n 2 = \frac{n (n + 1) (2 n + 1)}{6} l n 2 = l n 2^{\frac{n (n + 1) (2 n + 6)}{6}}$ $d^{'} o u p_{n} = 2^{\frac{n (n + 1) (2 n + 6)}{6}}$ $S_{n} = \underset{k = 1}{\sum^{n}} l n (u_{k}) = Σ k^{2} l n 2 = \frac{n (n + 1) (2 n + 1)}{6} l n 2$ $2 - a) v_{n + 1} - v_{n} = \sqrt{{l n u}_{n + 1}} - \sqrt{{l n u}_{n}} = \sqrt{{n + 1)}^{2} l n 2} - \sqrt{n^{2} l n 2} = (n + 1) \sqrt{l n 2} - n \sqrt{l n 2} = \sqrt{l n 2} d o n c (v_{n}) e s t u n e s u i t e a r i t h m e t i q u e d e r a i s o n \sqrt{l n 2}$ $n a t u r e o n a v_{k + 1} - v_{k} = \sqrt{} l n 2 \Rightarrow \underset{k = 1}{\sum^{n - 1}} (v_{k + 1} - v_{k}) = \sqrt{l n 2} \underset{k = 1}{\sum^{n - 1}} 1 \Leftrightarrow v_{n} - v_{1} = (n - 1) \sqrt{l n 2} e n f e s a n t t e n d r e l a l i m i t e o n v o i t q u e (v_{n}) e s t u n e s u i t e d i v e r g e n t e$
	Commented by RB95 last updated on 19/Sep/21
	Merci!
	Answered by Jonathanwaweh last updated on 19/Sep/21

	$b) S_{n} = Σ l n (v_{k}) = \underset{k = 1}{\sum^{n}} l n (\sqrt{{l n u}_{k}}) = Σ l n (\sqrt{l n (2^{k^{2}})}$ $= Σ l n (\sqrt{k^{2} l n 2})$ $= \underset{k = 1}{\sum^{n}} l n (k \sqrt{l n 2})$ $= \underset{k = 1}{\sum^{n}} l n k + \underset{k = 1}{\sum^{n}} l n (\sqrt{l n 2})) =$ $= l n (n!) + n l n (\sqrt{l n 2})$ $3 - a) p_{n} = Π t_{k} = Π {k e}^{- 2 k} o n a {l n p}_{n} = Σ l n ({k e}^{- 2 k})$ $= Σ l n k + l n (e^{- 2 k}) = Σ l n k - 2 k = n! - (n) (n + 1)$ $d^{'} o u p_{n} = e^{l n (n!) - (n) (n + 1)} = n! e^{- n (n + 1)}$ $b) \lim_{n \to + \infty} t_{n} = \lim_{n \to \infty} \frac{n}{e^{2 n}} = 0$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum