Asymptotics of Implicit Functions and Recurrent Sequences

Let $y(x)=\int_{x}^{+\infty } e^{-t^{2} / 2} d t(x \in \mathbb {R}), x(y)$ be the function inverse to $y(x)$. Prove that $x(y) \underset {y \rightarrow +0}{\sim } \sqrt{2 \ln \frac{1}{y}}$.

Prove that each of the following equations defines, in some neighborhood of the point $\left(x_{0}, c_{0}\right)$, an infinitely differentiable implicit function $y$, and find the first $n$ coefficients of the expansion $y(x)=c_{0}+c_{1}\left(x-x_{0}\right)+\ldots +c_{n}\left(x-x_{0}\right)^{n}+\ldots$

Verify that the domain of definition of the implicit functions considered in problem 4.2 contains the half-line $\left[x_{0},+\infty \right)$, and prove that as $x \rightarrow +\infty$

Prove that the equation $y^{x}+y=1$ defines, for all $x>0$, a unique implicit function $y$ and that

How many continuous solutions $y$ does the equation $x^{5}+x^{2} y=y^{3}$ have that tend to zero as $x \rightarrow 0$? What is their asymptotic behavior as $x \rightarrow 0$?

Prove that Kepler's equation $y=M+x \sin y$, where $M$ is a fixed positive parameter, defines in a neighborhood of the point $(0, M)$ an infinitely differentiable function $y$, and moreover

The problem of "the globe in a string bag." The Earth's globe was girdled with an inextensible thread. Then the thread was lengthened by 1 cm and pulled taut by tugging at the knot. Compute the height to which it rose, with a relative error of no more than one ten-thousandth of a percent (take the radius of the globe to be $6400\ \text{km}$).

Find the derivatives with respect to $\varepsilon$ of the lengths of the semi-axes of the ellipsoid $x^{2}+y^{2}+z^{2}+x y+y z+z x=1+\varepsilon x y$ at $\varepsilon =0$.

Let $x_{n}$ be the root of the equation $x=\operatorname {tg} x$ lying in the interval ($\pi n, \pi n+\pi$), $n \in \mathbb {N}$. Prove that

Let $x_{n}=h\left(x_{n-1}\right)$, where $h(x)=\frac{x}{\left(1+a x^{p}\right)^{1 / p}}$ for $x \geqslant 0$ ($a, p$ are positive constants, $n \in \mathbb {N}, x_{0}>0$). Prove that $x_{n} \sim (a n)^{-1 / p}$.

Let $x_{0}=10^{-3}, x_{n}=x_{n-1}-x_{n-1}^{2} \quad (n \in \mathbb {N})$. Prove that $x_{1000} \approx \frac{1}{2} 10^{-3}$ with accuracy $10^{-6}$.

Find the asymptotics of the recurrent sequences $x_{n}=f\left(x_{n-1}\right)\left(n \in \mathbb {N}, x_{0}>0\right)$ in the following cases:

Let $C, p, x_{0}>0$ and $x_{n}=x_{n-1}+C x_{n-1}^{1-p}(n \in \mathbb {N})$. Prove that

Construct a function $f$, positive on ($0,+\infty$), such that the sequence $x_{n}=f\left(x_{n-1}\right)\left(n \in \mathbb {N}, x_{0}>0\right)$ satisfies the relation $x_{n} \sim \frac{1}{\ln n}$.

Find the asymptotics of the recurrent sequences in the following cases ($n \in \mathbb {N}, x_{0}>0$):

Let $\quad x_{0} \in (-1,2), \quad x_{n}=x_{n-1}\left(x_{n-1}-1\right) \quad (n \in \mathbb {N})$. Find the asymptotics of the sequence $\left\{ x_{n}\right\}$.

Let $x_{0}=1, x_{n}=x_{n-1}+\left(\sum_{0 \leqslant k<n} x_{k}\right)^{-1}(n \in \mathbb {N})$. Prove that $x_{n} \sim \sqrt{2 \ln n}$.

Let $\left|p\right|<1, x_{0}=1, x_{n}=x_{n-1}+\left(\sum_{0 \leqslant k<n} x_{k}\right)^{-p}(n \in \mathbb {N})$. Find the asymptotics of the sequence $\left\{ x_{n}\right\}$.

Let $p \in \mathbb {R}, x_{0}=1, x_{n}=x_{n-1}+\left(\sum_{0 \leqslant k<n} x_{k}^{p}\right)^{-1}(n \in \mathbb {N})$. Find the asymptotics of the sequence $\left\{ x_{n}\right\}$.