find-nature-of-the-sequence-U-n-1-n-k-1-n-1-k-2-

Question Number 71664 by mathmax by abdo last updated on 18/Oct/19

f i n d n a t u r e o f t h e s e q u e n c e U_{n} = \frac{1}{n} {(\sum_{k = 1}^{n} \frac{1}{k})}^{2}

Commented by mathmax by abdo last updated on 18/Oct/19

w e h a v e \sum_{i = 1}^{n} x_{i}^{2} ⩾ \frac{1}{n} {(\sum_{i = 1}^{n} x_{i})}^{2} (r e s u l t p r o v e d)

l e t x_{i} = \frac{1}{i} \Rightarrow \sum_{i = 1}^{n} \frac{1}{i^{2}} ⩾ \frac{1}{n} {(\sum_{i = 1}^{i} \frac{1}{i})}^{2} \Rightarrow 0 < U_{n} ⩽ \sum_{i = 1}^{n} \frac{1}{i^{2}} \Rightarrow

0 ⩽ {l i m}_{n \to + \infty} U_{n} ⩽ \frac{π^{2}}{6} t h e s e q u e n c U_{n} c o n v e r g e s .

a n o t h e r w a y w e h a v e \sum_{k = 1}^{n} \frac{1}{k} = H_{n} = l n (n) + γ + o (\frac{1}{n}) \Rightarrow

\Rightarrow H_{n} \sim l n (n) + γ \Rightarrow H_{n}^{2} \sim {l n}^{2} (n) + 2 γ l n (n) + γ^{2} \sim {l n}^{2} (n) (n \to + \infty) \Rightarrow

U_{n} \sim \frac{{l n}^{2} (n)}{n} w e h a v e {l i m}_{n \to + \infty} \frac{{l n}^{2} (e^{n})}{e^{n}} = {l i m}_{n \to + \infty} n^{2} e^{- n} = 0 \Rightarrow

{l i m}_{n \to + \infty} U_{n} = 0

Answered by mind is power last updated on 18/Oct/19

\forall n \in N^{*} we have \sum_{k = 1}^{n} \frac{1}{k} ⩾ 1

\Rightarrow Un ⩾ \frac{1}{n}

\Rightarrow Σ U_{n} ⩾ Σ \frac{1}{n} \to + \infty

Commented by mathmax by abdo last updated on 18/Oct/19

s i r t h e q u e s t i o n i s f i n d t h e n a t u r e o f t h e s e q u e n c e n o t t h e s e r i e \dots

Commented by mind is power last updated on 18/Oct/19

ok sorry

for the sequence

\underset{k = 1}{\sum^{n}} \frac{1}{k} ∽ \ln (n)

\Rightarrow Un ∽ \frac{\ln^{2} (n)}{n} \to 0

un \to 0