Asymptotic calculus and equivalents

OmegaGauss

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Aug 25, 2019
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So, here's the exercise:
Let fn(x) = x^n +x-1 for n a natural number.

1) Prove that the equation fn(x)=0 has an unique positive solution (called xn)
2)Prove that (xn) converges to 1
3) Give a simple equivalent to (xn-1)

I've already done questions 1 and 2, which weren't all that hard, but the third one is giving me quite some trouble. I've tried finding an equivalent to ln(xn) (by applying ln to the equation), and I've tried to see if any simple function would do the trick, but it didn't seem to work. I then created the series yn=xn-1, and tried manipulating the equation I had, but I haven't gotten anything valuable out of it yet.

Any help would be appreciated!
 
Before I even start thinking about this, let's clarify what is being asked.

[MATH]n \in \mathbb Z^+ \implies f_n(x) = x^n + x - 1.[/MATH]
[MATH]\text {PROVE } \exists \text { a unique positive number } y_n \text { isuch that } f_n(y_n) = 0.[/MATH]
Is that correct? And you need no help with that?

[MATH]\text {PROVE } \lim_{n \rightarrow \infty} y_n = 1.[/MATH]
Is that correct? And you need no help with that either? And where you want help is

[MATH]y_n - 1 = \text {WHAT?}[/MATH]
Is that too correct?

Assuming all that is correct, what are your results for the first two questions? I have not so demonstrated, but it seems intuitive that the answer to the third question is highly dependent on the answers to the first two questions.
 
Actually, I suspect the question asks for a simplified form of \(x_{n-1}\)... though all of this confusion could really be avoided with a bit of proofreading and proper use of subscripts. Even x_(n-1) would be far clearer than what we have here.
 
" PROVE ∃ a unique positive number yn such that fn(yn)=0 "

Yes, that is correct, and I think I already have the answer to it. We just differenciate the function fn, show that it is positive, and as such, fn is an increasing function.
Then, we show that fn(0) = -1 and fn(1) = 1.
Finally, we can conclude with the intermediate value theorem, thus proving the existence and unicity of such a root for all n.

"PROVE [MATH]\lim_{n\to\infty}\\[3cm][/MATH]. yn = 1"

I have already managed to solve this one too; we can prove it by contradiction.
First, we make it clear that 0 < yn <1

Then, we prove that yn converges:
fn+1 (yn) =
gif.latex
+ yn -1 <
gif.latex
+ yn -1 = 0 = fn+1(yn+1)

Since fn is an increasing function, yn < yn+1 : (yn) is an increasing sequence, and is bounded; as such, yn converges.


Then, we suppose that yn doesn't converge towards 1; then ∃ a<1 such as yn < a for all n.
1 - a^n < yn = 1 -
gif.latex
< 1
Through the squeeze theorem, we see that [MATH]\lim_{n\to\infty}\\[3cm][/MATH]. yn = 1 , which goes against the hypothesis. Thus, by contradiction, [MATH]\lim_{n\to\infty}\\[3cm][/MATH]. yn = 1

For the third question, which is the one I need help with, it is not quite asked to simplify yn, but rather, to find the second term of the asymptotic expansion of yn (I know, I expressed myself really poorly).

Basically, I have to find a sequence unfor which
[MATH]\lim_{n\to\infty}\\[3cm][/MATH]. (yn-1)/un = 1

I think that now my post should be at least a little clearer.
 
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