Define the sequence [an]n≥2 by an=1+2+…+n It is well known an→l for a certain real number l. Show that limn→∞n⋅l−ann=e2

1) We want to estimate the difference l−xn. To do this note that l−xn=∑m≥n(xm+1−xm) So we find asymptotics for xm+1−xm 2) Show that xn+1−xn=n+1∏1≤k≤n(xn+1(k)+xn(k)} where xn(k)=k+(k+1)+…+n 3) Using xn(k)=k+xn(k+1) show that k≤xn(k)≤k+1 (3.1) xn(k)=k+12+O(1k) (3.2) xn(k)=kexp⁡(12k+O(1k32)) (3.3) for 1≤k&lt;n What is more, constant in O symbols are independent of k and n 4) Using 3.3 show that xn+1−xn=1+o(1)2n(n−1)!exp⁡(−∑1≤k&lt;n(12k+O(1k32))) hence xn+1−xn∼const2n(n−1)!exp⁡(−n) and l−xn∼xn+1−xn∼const2n(n−1)!exp⁡(−n) 5) Applying Stirlings

Solved: "Define the sequence [an]n≥2 by an=1+2+…+n It is..."

Tori Hines

Answered question

2022-01-22

Define the sequence

{[a_{n}]}_{n \geq 2}

a_{n} = \sqrt{1 + \sqrt{2 + \dots + \sqrt{n}}}

It is well known

a_{n} \to l

for a certain real number l. Show that

lim_{n \to \infty} \sqrt{n} \cdot \sqrt[n]{l - a_{n}} = \frac{\sqrt{e}}{2}

Answer & Explanation

trnovitom06

Beginner2022-01-23Added 12 answers

1) We want to estimate the difference

l - x_{n}

. To do this note that

l - x_{n} = \sum_{m \geq n} (x_{m + 1} - x_{m})

So we find asymptotics for

x_{m + 1} - x_{m}

2) Show that

x_{n + 1} - x_{n} = \frac{\sqrt{n + 1}}{\prod_{1 \leq k \leq n} (x_{n + 1}^{(k)} + x_{n}^{(k)}}}

where

x_{n}^{(k)} = \sqrt{k + \sqrt{(k + 1) + \dots + \sqrt{n}}}

3) Using

x_{n}^{(k)} = \sqrt{k + x_{n}^{(k + 1)}}

show that

\sqrt{k} \leq x_{n}^{(k)} \leq \sqrt{k} + 1

(3.1)

x_{n}^{(k)} = \sqrt{k} + \frac{1}{2} + O (\frac{1}{\sqrt{k}})

(3.2)

x_{n}^{(k)} = \sqrt{k} \exp (\frac{1}{2 \sqrt{k}} + O (\frac{1}{k^{\frac{3}{2}}}))

(3.3)
for

1 \leq k < n

What is more, constant in O symbols are independent of k and n
4) Using 3.3 show that

x_{n + 1} - x_{n} = \frac{1 + o (1)}{2^{n} \sqrt{(n - 1)!}} \exp (- \sum_{1 \leq k < n} (\frac{1}{2 \sqrt{k}} + O (\frac{1}{k^{\frac{3}{2}}})))

hence

x_{n + 1} - x_{n} \sim \frac{c o n s t}{2^{n} \sqrt{(n - 1)!}} \exp (- \sqrt{n})

and

l - x_{n} \sim x_{n + 1} - x_{n} \sim \frac{c o n s t}{2^{n} \sqrt{(n - 1)!}} \exp (- \sqrt{n})

5) Applying Stirlings

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Calculus 2

Didn't find what you were looking for?