Show that $a_n=\frac{\sqrt{n^{2}+2021n+420}}{\sqrt{n^{3}+2022n+420}}$ is decreasing

Question

I need to show that the sequence $a_n=\frac{\sqrt{n^{2}+2021n+420}}{\sqrt{n^{3}+2022n+420}}$ is decreasing.

I tried showing $a_{n+1}\le a_n$ but it was too messy.
I did manage to do it by showing that $f'(x)$ will be negative at some point by taking the limit of the nominator of the derivative to infinity, but this method was pretty exhausting. I wonder if there a simpler method I just didn't see, and i'd be happy to have it in my "toolbox".

thanks!

Answer 1

If $n\geq 1$, $$ \frac{{a_n }}{{a_{n + 1} }} = \sqrt {\frac{{n^5 + 2024n^4 + 8508n^3 + 4096228n^2 + 5787803n + 1026060}}{{n^5 + 2023n^4 + 4464n^3 + 4090926n^2 + 5787384n + 1025640}}} > 1, $$ since every term in the numerator is larger than the corresponding one in the denominator.

Answer 2

Alternative approach:

Since each element of $a_1, a_2, \cdots $ is positive, $a_1, a_2, \cdots $ is decreasing if and only if $(a_1)^2, (a_2)^2, \cdots$ is decreasing.

Let $b_n$ represent the numerator of $(a_n)^2$, and let $c_n$ represent the denominator of $(a_n)^2$. Then it is sufficient to show that beyond a finite number of leading terms, the fraction $\frac{c_n}{b_n}$ is increasing.

Personally, I favor polynomial long division here.

$$b_n(n - 2021) = c_n + D_1n + D_2$$

where $D_1, D_2$ are fixed constants.

Therefore, $\dfrac{c_n}{b_n}$ has the form

$$\left(n - 2021 - \frac{D_1n + D_2}{b_n}\right)\tag1$$

Clearly, there exists $N \in \Bbb{Z^+},$ such that for all $n\geq N, ~~\left|\dfrac{D_1n + D_2}{b_n}\right| < 1.$

Therefore, for $n \geq N,$ as $n \to (n+1)$, $(n - 2021)$ has increased by $1$ to $([n+1] - 2021)$, which must overshadow, any effect of the corresponding fraction,
$~\dfrac{D_1(n+1) + D_2}{b_{n+1}}.$

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Edit
Note that the question requires that it be shown that the sequence $\langle a_n\rangle$ is decreasing, rather than showing that the corresponding function is (beyond a certain point) strictly decreasing. This distinction allows my analysis.

Answer 3

$$a_n=\frac{\sqrt{n^{2}+2021n+420}}{\sqrt{n^{3}+2022n+420}} \implies a_n^2=\frac {n^{2}+2021n+420}{n^{3}+2022n+420 }$$

Long division or Taylor series $$a_n^2 =\frac{1}{n}+\frac{2021}{n^2}-\frac{1602}{n^3}+O\left(\frac{1}{n^4}\right)$$ Doing the same $$a_{n+1}^2=\frac{1}{n}+\frac{2020}{n^2}-\frac{5643}{n^3}+O\left(\frac{1}{n^4}\right)$$ $$\frac{a_{n+1}^2 } {a_n^2}=\frac{\frac{1}{n}+\frac{2020}{n^2}-\frac{5643}{n^3}+O\left(\frac{1}{n^4}\right) } {\frac{1}{n}+\frac{2021}{n^2}-\frac{1602}{n^3}+O\left(\frac{1}{n^4}\right) }$$ Long division again $$\frac{a_{n+1}^2 } {a_n^2}=1-\frac{1}{n}-\frac{2020}{n^2}+O\left(\frac{1}{n^3}\right)= 1-\frac{1}{n}+O\left(\frac{1}{n^2}\right)$$ $$\frac{a_{n+1} } {a_n}=1-\frac{1}{2n}+O\left(\frac{1}{n^2}\right)$$

Answer 4

Thank you for your answers, as there were no magic shortcuts, I thought i'd share my original way.
I will say though that for my case it was enough to show $a_n$ is decreasing at some point.

We define $f\left(x\right)=\sqrt{\frac{x^{2}+2021x+420}{x^{3}+2022x+420}},\;\;\forall x\ge 1$. by deriving we get:

$$f'\left(x\right)=\frac{\left(2x+2021\right)\left(x^{3}+2022x+420\right)-\left(x^{2}+2021x+420\right)\left(3x^{2}+2022\right)}{2f\left(x\right)\left(x^{3}+2022x+420\right)^{2}}$$

While its clear that both $f$ and the denominator are always positive, we check the limit of the nominator: $$\lim_{x\to\infty}x^{4}\left[\left(2+\frac{2021}{x}\right)\left(1+\frac{2022}{x^{2}}+\frac{420}{x^{3}}\right)-\left(1+\frac{2021}{x}+\frac{420}{x^{2}}\right)\left(3+\frac{2022}{x^{2}}\right)\right]=\infty\left[2-3\right]=-\infty$$

Hence there exists $ N\in\mathbb{N}$ such that $f'(x)<0\;\;\forall x\ge N$, thus $\left(a_{n}\right)_{n=N}^{\infty}$ is decreasing.

Answer 5

"I tried showing an+1≤an but it was too messy. "

No it's not.

$a_{n+1} \le a_n \iff $

$\frac{\sqrt{(n+1)^{2}+2021(n+1)+420}}{\sqrt{(n+1)^{3}+2022(n+1)+420}} \le \frac{\sqrt{n^{2}+2021n+420}}{\sqrt{n^{3}+2022n+420}}\iff$

$\sqrt{n^{3}+2022n+420}\sqrt{(n+1)^{2}+2021(n+1)+420}\le \sqrt{n^{2}+2021n+420}\sqrt{(n+1)^{3}+2022(n+1)+420}\iff$

$(n^{3}+2022n+420)((n+1)^{2}+2021(n+1)+420)\le (n^{2}+2021n+420)((n+1)^{3}+2022(n+1)+420)$

$(n^3 + 2022n + 420)(n^2 + 2023n + 2422) \le (n^2 + 2021n + 420)(n^3 +3n^2 + 2025n +2443) \iff$

$n^5 + 2023n^4+ (2022+2422)n^3 + (420 + 2022*2023)n^2 +(2022*2422+2023*420)n + 420*2422 \le n^5 + (3+2021)n^4 + (420+3*2021+2025)n^3 + (420*3 + 2021*2025+ 2443)n^2 + (2021*2443 + 420*2025)n + 420*2443\iff$

$n^5 + \color{red}{2023}n^4+ \color{green}{(2022+2422)}n^3 + \color{blue}{(420 + 2022*2023)}n^2 +\color{purple}{(2022*2422+2023*420)}n + 420*2422 \le n^5 \color{red}{(3+2021)}n^4 + \color{green}{(420+3*2021+2025)}n^3 + \color{blue}{(420*3 + 2021*2025+ 2443)}n^2 + \color{purple}{(2021*2443 + 420*2025)}n + 420*2443$

ANd as all the terms on the LHS are distinctly less than or equal to the corresponding terms on the RHS that is clearly true