All Questions
12
questions
2
votes
1
answer
659
views
What function does $\sum_{n=0}^\infty \frac{(-1)^n(2n)!}{(1-2n)n!^24^n}\frac{x^n}{y^{2n}}$ correspond to?
I have the following Taylor series, and although it looks familiar, I cannot figure out which function it corresponds to! Does anyone recognize this Taylor series?
$$1+\frac{x}{2y^2} - \frac{x^2}{8y^...
- 149
1
vote
1
answer
479
views
Taylor series for $e^{-x \ln x}$
What is a Taylor expansion for the following function?
$$ e^{-x \ln x} $$
I assume you can't do a Taylor expansion around $x=0$, since the function doesn't exist at that point. The next best choice ...
- 161
3
votes
1
answer
99
views
Is there a closed expression for this one?
Has anyone ever seen a closed expression (and its sequence) for the following one:
$$\partial_x^{n} \, \text{arctanh}^k(x)|_{x=0}\,\,\,\,(1)$$
Provided $k,n \in \mathbb{N_0}$ and $x \in \mathbb{R}$.
...
- 516
-1
votes
2
answers
809
views
closed form of Taylor series expansion [closed]
Is there a closed form of this summation?
$A = \sum_{j=0}^\infty \frac{1}{j!}\times \frac{1}{j!}x^j$
or can it be derived as multiple of two function? ( ex) $A = \cos x\times e^x$ )
- 69
0
votes
2
answers
157
views
Taylor Series about the function $\log(x + 1)$
I read the following taylor expresion for
$$\log (x+1)=\sum _{k=1}^n \frac{(-1)^{k+1} x^k}{k}+\frac{(-1)^n x^{n+1} \, _2F_1(1,n+1;n+2;-x)}{n+1}$$
Do you know where it comes from?
user167276
1
vote
1
answer
238
views
Taylor series expansion of $(x^2-x)\ln(1-x)$ and calculating a sum
So I have this function: $$f(x)=(x^2-x)\ln(1-x)$$
So I want to calculate it's Taylor series centered at x=0, basically that is Maclaurin series, and that series will be of help when calculating this ...
- 2,091
2
votes
2
answers
286
views
Find a function for the infinite sum $\sum_{n=0}^\infty \frac{n}{n+1}x^n$
I need to find a function $f(x)$ which is equal to the sum
$$
\sum_{n=0}^\infty \frac{n}{n+1}x^n,
$$
for every $x\in \mathbb{R}$ for which the series converge.
Now, using WolframAlpha, I've found the ...
- 435
1
vote
1
answer
36
views
How to show $\sum\limits_{i=1}^{t}\frac{1}{i}2^{t-i}=2^t\ln 2 -\frac{1}{2}\sum\limits_{k=0}^\infty \frac{1}{2^k(k+t+1)}$
How to show the below equation ?
$$\sum\limits_{i=1}^{t}\frac{1}{i}2^{t-i}=2^t\ln 2 -\frac{1}{2}\sum\limits_{k=0}^\infty \frac{1}{2^k(k+t+1)}
~~~~~(t\in \mathbb Z^+)$$
- 1,543
5
votes
2
answers
283
views
Closed formula for the asymptotic limit of a definite integral
I would like to solve the following integral:
$$ I_0 (a,b)= \int_0^1 dx\int_0^{1-x} dz \frac{1}{a z (z-1)+a x z + x(1-b)}$$
in the limit where $b$ is small ($a$ and $b$ are positive constants).
...
- 87
4
votes
2
answers
215
views
Evaluating $\lim_{x \to 0}\frac{(1+x)^{1/x} - e}{x}$ [duplicate]
How to evaluate the following limit? $$\lim_{x \to 0}\frac{(1+x)^{1/x} - e}{x}.$$
- 1,301
8
votes
5
answers
749
views
Closed-forms of infinite series with factorial in the denominator
How to evaluate the closed-forms of series
\begin{equation}
1)\,\, \sum_{n=0}^\infty\frac{1}{(3n)!}\qquad\left|\qquad2)\,\, \sum_{n=0}^\infty\frac{1}{(3n+1)!}\qquad\right|\qquad3)\,\, \sum_{n=0}^\...
- 19.4k
4
votes
2
answers
244
views
How to find a closed form for the derivatives of $F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,dt,$ $F(0)=\frac12$?
I have been given the function $$F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,{\rm d}t$$ for $x\ne 0,$ $F(0)=\frac12,$ and charged with finding a Taylor polynomial for $F(x)$ differing from $F$ by no ...
- 104k