The objective of this exercise is to calculate lim_(n→+∞) (1/(√n)) ×Σ_(k=1) ^n (1/(√(n+k))). Given S_n =Σ_(k=1) ^n (1/(√k)); U_n =2(√2)−S_n and V_n =2(√(n+1))−Sn. 1. show thatlim_(n→+∞) Sn=+∞. 2. show that V_n and U_n are adjacent then deduct that their common limit is L≥1. 3. Calculate lim_(n→+∞) ((S_n /n)) and lim_(n→+∞) ((S_n /(√n))). 4. Deduct from last questions lim_(n→+∞) (1/(√n)) ×Σ


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Question Number 127430 by mathocean1 last updated on 29/Dec/20

$T h e o b j e c t i v e o f t h i s e x e r c i s e$ $i s t o c a l c u l a t e \underset{n \to + \infty}{l i m} \frac{1}{\sqrt{n}} \times \underset{k = 1}{\sum^{n}} \frac{1}{\sqrt{n + k}} .$ $G i v e n S_{n} = \underset{k = 1}{\sum^{n}} \frac{1}{\sqrt{k}}; U_{n} = 2 \sqrt{2} - S_{n}$ $a n d V_{n} = 2 \sqrt{n + 1} - S n .$ $1 . s h o w t h a t \underset{n \to + \infty}{l i m S n} = + \infty .$ $2 . s h o w t h a t V_{n} a n d U_{n} a r e$ $a d j a c e n t t h e n d e d u c t t h a t t h e i r$ $c o m m o n l i m i t i s L ⩾ 1 .$ $3 . C a l c u l a t e \underset{n \to + \infty}{l i m} (\frac{S_{n}}{n}) a n d$ $\underset{n \to + \infty}{l i m} (\frac{S_{n}}{\sqrt{n}}) .$ $4 . D e d u c t f r o m l a s t q u e s t i o n s$ $\underset{n \to + \infty}{l i m} \frac{1}{\sqrt{n}} \times \underset{k = 1}{\sum^{n}} \frac{1}{\sqrt{n + k}} .$
Commented by mathocean1 last updated on 29/Dec/20

$s o r r y$
Commented by talminator2856791 last updated on 29/Dec/20

$h a h a h a a a$
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