All Questions
Tagged with taylor-series polynomials
7 questions
78
votes
7
answers
8k
views
Roots of truncations of $ e^x - 1$
During a talk I was at today, the speaker mentioned that if you truncate the Taylor series for $e^x - 1$, you'll get lots of roots with nonzero real part, even though the full Taylor series only has ...
- 2,284
8
votes
1
answer
2k
views
Polynomial approximation for square root function with fast convergence and bounded coefficients
Let $\delta, \varepsilon \in (0,1)$. I am interested in a sequence $\{f_n\}$ of polynomial approximations of the square root function $x \to x^{1/2}$ on $[\delta,1]$, of the form
$$
f_n(x) = \sum_{i=0}...
- 483
2
votes
1
answer
136
views
Expressing a vector valued function in terms of its derivatives
Consider a function
$$
f:\mathbb{R}^n\rightarrow\mathbb{R}^m
$$
given by $m$ functions $f_i:\mathbb{R}^n\rightarrow \mathbb{R}$ that we can assume to be polynomials in $x_1,\dots,x_n$.
Does there ...
- 23
1
vote
0
answers
72
views
Approximation Rates for Multivariate Taylor Series
Let $k,n,m$ be positive integers and suppose that $f$ is $C^{k}(\mathbb{R}^n,\mathbb{R}^m)$ functions. For any given $\epsilon>0$ and $x_0\in \mathbb{R}^n$, are there known sharp approximation ...
- 5,405
1
vote
0
answers
121
views
Properties and name of some polynomials
I have encountered in a problem some polynomials given by $P_k(x) = \prod_{j=0}^{k-2} (kx-j)$. I need to understand if these polynomials are known, and if they have certain special properties, as ...
- 321
0
votes
1
answer
662
views
A polynomial and its reciprocal expansion [closed]
Suppose $f(x)=\prod_{k=1}^n(x-a_k)$ where all $a_k>0$.
Expand the function $\frac1f$ at $\infty$ so that
$$\frac1{f(x)}=\frac{b_n}{x^n}+\frac{b_{n+1}}{x^{n+1}}+\cdots.$$
Does it follow that each $...
- 309
-2
votes
0
answers
43
views
About the uniqueness of the Taylor polynomial
I'm in trouble understanding this theorem:
For a given function, differentiable n times at a given point $x_0$, there exist a unique polynomial $P_n$ (of degree $\le$n) such that
$$\forall \; k=0,...,...
- 1