All Questions
Tagged with power-series taylor-series
14 questions
30
votes
2
answers
17k
views
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{...
- 579
8
votes
3
answers
759
views
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 ...
- 10.1k
6
votes
2
answers
404
views
What's the summation of formal series $\sum_{n\geq0}\binom{n\delta}{n}x^n$？
$\delta$ is a positive number. Is this Taylor expansion of some function?
- 61
5
votes
2
answers
671
views
Lower Bounds for the Roots of Polynomials
I'm interested in the "size" of the roots of a sequence of Taylor Polynomials of an entire function.
For example, consider $\mathrm f(z) = \mathrm e^z$. The Taylor Polynomials, or $k$-jets, are
$$\...
- 310
5
votes
1
answer
384
views
Asymptotic growth of the of Taylor coefficients of the inverse of a function
Let $f(x)=\sum_{n\geq 1} c_n\cdot x^n$ be a function given by a power series. Further there is some $\alpha >1$ such that for all $n$, $c_n = \Theta(1/n^{\alpha})$. What can one say about the ...
- 61
4
votes
3
answers
698
views
What or where is the series expansion of the function $\ln\bigl(\frac{\tan x}{x}-1\bigr)$ or $\ln(\tan x-x)$ around $x=0$?
It is known that
\begin{equation*}
\tan x=\sum_{k=1}^{\infty}\frac{2^{2k}\bigl(2^{2k}-1\bigr)}{(2k)!}|B_{2k}|x^{2k-1}, \quad |x|<\frac{\pi}{2}
\end{equation*}
and
\begin{equation*}
\ln\tan x=\ln x+\...
- 1,101
4
votes
3
answers
506
views
An apparently simple question (behaviour at infinity of a power series)
Let $(a_n)$ be a sequence of real numbers, and suppose that the real power series (function) $S(x):=\sum_{n=0}^{\infty} a_n x^n$ converges for every $x\in\mathbb{R}$.
$\mathbf{Question}$: Suppose ...
- 51
3
votes
1
answer
81
views
Exponential taylor series for multiple variables with linear constraints for coefficients
I'm trying to simplify the sum
$$
\sum_{\vec x \in (\mathbb{N}_0)^n: M\vec x = \vec b} \prod_i \frac{(a_i)^{x_i}}{x_i!},
$$
where $M$ is a $\mathbb{N}_0$-valued $m\times n$ matrix, $\vec b$ is $\...
- 3,001
3
votes
0
answers
256
views
derivatives of composite function [closed]
There's a formula for the $n$th derivative of a composite function $f(g(x))$ - it's called Faa di Bruno's formula - but I'm not really interested in the formula but in the proof given in the book of ...
- 171
2
votes
1
answer
2k
views
construct a power series with infinitely many zeros in the complex plane, bounded coefficients???
Hi all.
I want to construct a power series $F(z)=\sum_{n=0}^\infty c_nz^n$ centered at zero and with finite radius of convergence in the complex plane, and which has infinitely many zeros (in its ...
- 1,591
1
vote
0
answers
35
views
Exponential-like function equivalent for the Dixonian Elliptics
Is there some exponential-like function that acts as partner for the intricate Dixonian Elliptics, in a similar way that the Exponential function acts as a partner for the trigonometric functions ?
1
vote
0
answers
52
views
A problem on monotonicity rule for the ratio of two Maclaurin power series
In the paper [1] below, a monotonicity rule for the ratio of two Maclaurin power series was presented as follow.
Let $a_k$ and $b_k$ for $k\in\{0\}\cup\mathbb{N}$ be real numbers and
the power series ...
- 1,101
1
vote
0
answers
196
views
Is a mixture of real analytic functions again analytic?
Let $$h : \mathbb{R}^2 \to \mathbb{R}^+.$$
Suppose that for each $x$, $h(x, y)$ is a real analytic function of $y$.
Let $\mu(dx)$ be a finite measure on $\mathbb{R}$, and for each $y$, suppose that
$$...
- 103
-4
votes
1
answer
197
views
How to express a quadratic polynomial exactly as a power series [closed]
I claim, for $\operatorname{artanh}(\rho) = \frac{1}{2} \ln\left(\frac{1+\rho}{1-\rho}\right)$, i.e., the inverse hyperbolic tangent function, the following holds approximately under assumptions given ...
- 191