Questions tagged [taylor-series]
Taylor series is a method to analyze functions as polynomials.
14 questions
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power series of the reciprocal... does a recursive formula exist for the coefficients [closed]
Let $f(x)=\sum _{n=0}^{\infty } b_nx^n$ and $\frac{1}{f(x)}=\sum _{n=0}^{\infty } d_nx^n$. Then the coefficients of the reciprocal of $f(x)$ can be written down. The first few terms are:
$d_0 = \frac{...
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78
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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 ...
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What's an example of a function whose Taylor series converges to the wrong thing?
Can anyone provide an example of a real-valued function f with a convergent Taylor series that converges to a function that is not equal to f (not even locally)?
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Transformation converting power series to Bernoulli polynomial series
I wonder, can anyone describe an expression or formula of a transform that converts
$$\sum_{k=0}^\infty \frac{a_k x^k}{k!}$$
into
$$\sum_{k=0}^\infty \frac{a_k B_k(x)}{k!}$$
where $B_k(x)$ are ...
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What or where is the series expansion of the function $\ln\bigl(\frac{\tan x}{x}-1\bigr)$ or $\ln(\tan x-x)$ around $x=0$?
It is known that
\begin{equation*}
\tan x=\sum_{k=1}^{\infty}\frac{2^{2k}\bigl(2^{2k}-1\bigr)}{(2k)!}|B_{2k}|x^{2k-1}, \quad |x|<\frac{\pi}{2}
\end{equation*}
and
\begin{equation*}
\ln\tan x=\ln x+\...
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Integral representation of higher order derivatives
I'm quite curious about the following phenomena, that still puzzle me although I have a proof, and I'd be really glad if someone may shred some light, showing an interpretation or a generalization. I ...
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What characterizes rational functions with nonnegative integer Taylor coefficients?
I believe that there is a statement along the following lines (I would, of course, love to be corrected): a formal power series is the Taylor expansion of a rational function if and only if the ...
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Taylor $k$-differentiability of a real function at a point
I am interested in the standard name for the following weak form of $k$-differentiability.
Definition. A function $f:\mathbb R\to\mathbb R$ is called Taylor $k$-differentiable at a point $x_0$ if ...
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The Basel problem revisited?
In the Basel problem, the $sinc$ function is considered at the Wikipedia page.
Let me try to make an alternative function definition:
$$f(x) = \prod_{n=1}^\infty \left ( 1+ \frac{x^3}{n^3} \right ) = \...
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Approximate the square root of (1-X) efficiently through (nested) products
Currently, I encountered a problem of approximating the following
series:
$$
(I-X)^{-\frac{1}{2}}=I+\frac{1}{2}X+\frac{1\cdot3}{2\cdot4}X^{2}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6}X^{3}+\ldots
$$
where ...
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One-Sided Analyticity Condition Guarantees Analytic Function?
Let $f \ \colon \ [0,\infty) \to \mathbb{R}$ be a function satisfying:
$f$ is differentiable infinitely many times in $(0,\infty)$, and has a right-derivative of any order at $0$.
$f$ satifsfies the ...
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Smoothness of coefficients of remainder term in Taylor expansion
Given a $C^{k}$ function $f:\mathbb{R}^d\to\mathbb{R},$ we can use Taylor's theorem to write it as
$$f(x)=\sum_{|\alpha|\le k-1} c_\alpha x^\alpha + R(x),$$
where $R$ is $C^k$ and can be expressed ...
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Show that the absolute value of this function is twice differentiable except on a set of Lebesgue measure $0$
Let
$f\in C^3(\mathbb R)$ with $f>0$ and $$\int f(x)\:{\rm d}x=1\tag1$$
$g:=\ln f$ and assume that $g'=\frac{f'}f$ is Lipschitz continuous (note that this implies that $f'(x)\xrightarrow{|x|\to\...
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Taylor Series expansion for an implicitely defined family of functions
Can we find a Taylor Series expansion for $y(x)$ implicitly defined by:
$$\sum _{i=1}^nA_ie^{a_ix+b_iy} = 1 ?$$
In financial mathematics, the two-additive-factors Model G2++ is commonly used for ...
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