All Questions
Tagged with taylor-series sequences-and-series
18 questions
3
votes
1
answer
212
views
Other expansion for positive Taylor expansion
I was thinking of the following problem. Let $f$ be a Taylor expansion and $a_k$ the associated coefficients,
$$\forall x\in\mathbb{R},~f(x)\triangleq\sum_{k=0}^\infty a_kx^k.$$
Let suppose that we ...
- 393
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
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
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
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
6
votes
0
answers
2k
views
Do smooth cutoff functions analytically continue functions?
My goal is to prove (or disprove) that sufficiently smooth and quickly decaying cutoff functions being tacked on to a Taylor series correctly extend the radius of convergence to the analytic ...
- 1,740
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
3
votes
2
answers
607
views
Are there any techniques that can be used in the case when a Neumann series doesn't converge?
Suppose we have a bounded linear operator $A = A(\gamma):H_1\to H_2$ where $H_1$ and $H_2$ are Hilbert spaces and $\gamma>0$ is some parameter, and we are interested in the solution to
$$
(I-A)x = ...
- 175
18
votes
3
answers
1k
views
A curious series related to the asymptotic behavior of the tetration
The tetration is denoted $^n a$, where $a$ is called the base and $n$ is called the height, and is defined for $n\in\mathbb N\cup\{-1,\,0\}$ by the recurrence
$$
{^{-1} a} = 0, \quad {^{n+1} a} = a^{\...
- 6,829
2
votes
3
answers
379
views
Compute inverse series for implicit equation $b=-\log(1-e^{-x})/x$
In financial mathematics, the inverse series of: $$b(x) = -\frac{\log(1-e^{-x})}{x}$$ is needed in order to perform fast calculation on swaptions for G2++ calibration model. (see this post for ...
- 181
1
vote
0
answers
56
views
Hadamard-like product on infinitely differentiable functions
Has the following operation $*$ on formal power series $f,g$ been studied before?
$$[X^n](f*g) = n! \cdot [X^n]f \cdot [X^n]g$$
where $n$ is a nonnegative integer? This is the typical Hadamard ...
- 11
3
votes
2
answers
247
views
aproximate sum involving binomial coefficients
I have the problem for computing the j-derivative of a logarithm, with $j\gg1$
\begin{equation}
c_j=\left.\frac{\partial^j}{\partial s^j}\log\left(1+Ae^s+Be^{2s}\right)\right|_{s=0},
\end{equation}
...
- 183
1
vote
0
answers
385
views
Generating a series representation for the inverse of the operator $f(f)$
I am considering the following problem:
Suppose you are given a function $u: C \rightarrow C$, find a function $g$ such that $g(g) = u$ (Let's assume that such a function exists). And by "find", I ...
- 2,689
1
vote
1
answer
154
views
A nonlinear initial-boundary value problems with Taylor expansion of parameter [closed]
Let $u(x,t; \epsilon)$ satisfy the nonlinear initial boundary value problem
$$
u_{tt} = (u_{x} + u_{x}^3)_{x} + u_{xxt}, \space 0 \lt x \lt 1
$$
$$
u(0,t) = 0 \\
u(1,t) = 0 \\
u(x,0) = \epsilon f(x) \...
- 111
2
votes
1
answer
919
views
Series approximation(s) of a difficult recursive equation
New user here. I'm working on trying to get asymptotic solutions to the following recursive function:
$f(r)=\frac{1}{r-k}\lgroup\sqrt{\frac{2}{k^2-1}}+\sqrt{\frac{1}{2k^2-1}}\rgroup$ (Eqn. 1)
...
- 121
1
vote
0
answers
499
views
Applying the ideas of power series to certain convolutions - which identities transfer?
Let's suppose I'm working with some set of functions $f_k(n)$. $f_1(n)$ is essentially the root of my functions, and could be nearly anything, and then $f_k(n) = (f_1(n) * f_{k-1}(n))$ for some ...
- 627
0
votes
0
answers
173
views
Series expansion with remaining $log n$
Hi,
I'm studying the asymptotic behavior $(n \rightarrow \infty)$ of the following formula, where $k$ is a given constant.
$$ \frac{1}{n^{k(k+1)/(2n)}(2kn−k(1+k) \ln n)^2}$$
I'm trying to do a ...
- 83