All Questions
10
questions with no upvoted or accepted answers
2
votes
0
answers
67
views
Closed form for $\psi^{1/k}(1)$, where $k$ is an integer
I have proven the identity
$$
\sum_{k=1}^{\infty} \dfrac{\operatorname{_2F_1}(1, 2, 2-1/t,-1/k)}{{k}^{2}} = Γ(2-\dfrac{1}t){\psi^{1/t}(1)}+\psi(-\dfrac{1}t)(\dfrac{1}t(1-\dfrac{1}t))+\gamma(1-\dfrac{1}...
2
votes
0
answers
93
views
N'th derivative of $f(x)$ to the power of $g(x)$
I am trying to find a formula for the $n$'th derivative of $f(x)^{g(x)}$. I have tried using the formula bellow along with Leibniz rule without success.
$$D^n(f(g(x)))=\sum_{k=0}^{n-1} \binom {n-1}{k}...
- 436
1
vote
0
answers
57
views
How to approximate $\prod\limits_{(i, j)\in I^2} (1 - \alpha_i x_{i, j})\;$ as a linear function?
Suppose we have a finite set $I \subseteq \mathbb{N}$, and $\alpha_i\in [0, 1]$ are fixed numbers for all $i\in I$.
Is there a way to approximate $\prod\limits_{(i, j)\in I^2} (1 - \alpha_i x_{i, j})...
- 342
1
vote
0
answers
62
views
What is the the function $f(x)$ has the property $\sum_{n=1}^x f(x^n)=x^2$
What is the the function $f(z)$ has the property $\sum_{n=0}^z f(z^n)=z^2$ when $z$ is a natural number bigger than $0$, and the function $f(z)$ is differentiable everywhere.
$$f(1)+f(1)=1$$
$$f(1)+f(...
user913631
1
vote
0
answers
75
views
Taylor Series inverse function
Find a function represented by the Taylor series $\displaystyle\sum_{k=0}^{\infty}(-1)^k\cdot\dfrac{3^{2k+1}}{(2k+1)!}\cdot x^{2k}$.
This series looks very similar to the result for sin 3x. However, ...
- 11
1
vote
1
answer
42
views
What is the series of the function 3 / ( 1- x^4)
I know that $f(x) = \frac{1}{ 1-x } = \sum_{n=1}^\infty x^n$. We can find that $g(x) = \frac{1}{ 1-x^4 } = \sum_{n=1}^\infty (x^4)^n = \sum_{n=1}^\infty x^{4n}$. Does the sum converge? what is the ...
- 11
0
votes
1
answer
61
views
Cool identities/properties involving the Alternating Harmonic Numbers
Using the following analytic continuation for the Alternating Harmonic Numbers ($\bar{H}_x=\sum_{i=1}^x\frac{(-1)^{i+1}}i$): $$\bar{H}_x=\ln2+\cos(\pi x)\left(\psi(x)-\psi\left(\frac x2\right)-\frac1x-...
- 6,549
0
votes
1
answer
59
views
Integral from infinite sum
I am trying to find if there is a way of turning the following sum into an integral:
$$\lim_{x\to\infty} \sum_{k=1}^{x}\operatorname{tr}\left(\frac{C^{k}}{x^2+x}\right)$$
I have looked into Riemann ...
- 241
0
votes
0
answers
85
views
Proving $\sum_{k=1}^na_k\sin(kx)=\sum_{k=1}^nb_k\cos(k x)$, with real $a_k$ and $b_k$, has at least one real solution
Prove that for every positive integer $n$ and any real numbers $a_1,a_2,\cdots , a_n, b_1,\cdots, b_n $, the equation
\begin{equation} \phantom{=}a_1 \sin(x)+a_2 \sin(2x)+\cdots +a_n \sin(nx) \\ = b_1 ...
- 155
0
votes
0
answers
100
views
Find the function $f(x)$ under the given conditions
I need to find $f(x)$,
If $$ \frac{1/x_m}{\sum_u 1/x_u} = \frac{f(x_m)}{\sum_u f(x_u)} $$
$x_u > 0, \forall u$, $u = 0,1,2,3,...,U$, and $x_m = x_u$ if $m = u$
Could $f(x)$ be any thing other ...
- 13