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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