All Questions
Tagged with mathematica sequences-and-series
23
questions
4
votes
0
answers
144
views
Analysis of a sequence based on ratios of floors of consecutive powers of a real value
I am currently interested in the following sequence:
Fix a number $x$, and define a sequence $(a_n)_{n=0}^\infty = \lfloor x^n \rfloor$ and $(b_n)_{n=0}^\infty = \frac{a_{n+1}}{a_n}$
With a little ...
- 411
3
votes
0
answers
306
views
Is Wolfram Alpha giving me a wrong answer?
I have asked for the convergence of the series $\sum (3^n/\sqrt{n})x^{2n+1}$, which has the radius of convergence of $1/\sqrt{3}$ and diverges at $|x|=1/\sqrt{3}$. However, the Wolfram Alpha is ...
- 560
1
vote
1
answer
129
views
Algorithm used by Mathematica for evaluating partial sums
Today, while using mathematica, I found that:
$$\sum_{x\geq n\geq0}\frac{1}{n!}=\frac{e\Gamma(x+1,1)}{\Gamma(x+1)}$$
Where $\Gamma(x,y)$ is the Incomplete Gamma function.
It is well known that
$$\sum_{...
user840532
2
votes
0
answers
61
views
Unique values for sums of form $1 + a_1 + a_1a_2 + \ldots + a_1\cdots a_k$?
Suppose you have some finite set $S$ of positive integers, and a finite sequence $A$ which can only use values in $S$. I am curious about whether you can specify $S$ such that no two sequences will ...
- 6,014
3
votes
1
answer
112
views
The infinite sum $\sum_{n=1}^\infty (-1)^{n+1} \frac{2n-1}{n^2-n+1}$
My (rather old) version of Mathematica cannot compute
$$\sum_{n=0}^\infty (-1)^{n+1}\frac{2n-1}{n^2-n+1}$$
other than re-writing it as a hypergeometric function as follows:
$$\mbox{...
- 1,136
1
vote
1
answer
88
views
Proving $ \sum^{\infty}_{k=1}\frac{\textrm{sin}(\frac{k\pi}{2})\textrm{cos}(k\delta)}{k}=1/4({\pi}+gd(i\delta)+gd(-i\delta))$
In Mathematica, we can show that
\begin{equation}\label{gd}
\sum^{\infty}_{k=1}\frac{\textrm{sin}(\frac{k\pi}{2})\textrm{cos}(kx)}{k}=\frac{1}{4}\big({\pi}+\textrm{gd}(ix)+\textrm{gd}(-ix)\big),
\end{...
- 343
3
votes
2
answers
436
views
What is the sum of series??
What is the sum of first 50 terms of the series
$$(1\times3)+(3\times5)+(5\times7)+\ldots$$
I had tried to solve this question
It seems that it is mixed arithmetic series
In which first $A.p$ is $1,...
- 91
0
votes
2
answers
69
views
How to find arithmetic mean of number [closed]
The arithmetic mean of 1,8,27,64... upto n terms is given by ....
I know the formula for arithmetic mean
But i don't know how to apply it
I applied a+b/2
But it's not logical here so
Plzz tell me ...
- 91
2
votes
1
answer
260
views
Series related to the (lower) incomplete gamma function $\gamma(a,x)$ with $x<0$
In a problem involving the (lower) incomplete gamma function $\gamma(a,x)$, with $a>0$ and $x<0$, I've decided to apply the following equivalence:
$$
\gamma(a,x) = e^{-x}x^a \sum_{k=0}^{+\infty}{...
- 241
0
votes
3
answers
117
views
series of integrals Mathematica Wolfram [closed]
Does somebody knows how to represent series of integrals in Mathematica? I would like to represent something like $I_n=\int_0^{\pi/4}\tan^{2n}xdx$ and prove convergence of it, but after searching docs,...
- 202
1
vote
1
answer
236
views
Sum of an exponential function/Mathematica
I have a function that looks like following:
$$T=2\sum_{n=1}^{\infty}\frac{(-1)^{n-1}\cos(\lambda_n x)}{\lambda_n}\exp(-\lambda_n^2 t)$$
where $\lambda_n=\frac{(2n-1)\pi}{2}$. I use Mathematica to ...
- 11
0
votes
1
answer
288
views
The Lerch Phi Function and a possible Mathematica Bug [Solved]
The Lerch Phi Function is defined as
$$\Phi(-s, \alpha, \nu) = \sum_{k = 0}^{+\infty} \frac{(-s)^k}{(k+\nu)^{\alpha}}$$
Now, in my special case I have $\alpha = 1$, hence it does simply reduce to
$$...
- 26.3k
0
votes
3
answers
1k
views
Calculate taylor series in Wolfram Mathematica or Maple
I want to calc the Taylor Series Expansion of $Â(z) = z\sum_{j\ge0}(e^z - e^{(1 - 2^{-j})z})$ at $z=0$, but I can't figure out a way to handle the infinite summation.
When I ask Maple
...
- 165
3
votes
3
answers
309
views
How can I calculate OEIS A144311 efficiently?
I'm looking for a way to calculate OEIS A144311 efficiently.
In one sense or another, this series considers the number between "relative" twin primes. What do I mean by this?
Well, the number $77$ ...
11
votes
1
answer
3k
views
Fractional oblongs in unit square via the Paulhus packing technique
Oblongs of size $ \frac{1}{1} \times \frac{1}{2}$, $ \frac{1}{2} \times \frac{1}{3}$, $ \frac{1}{3} \times \frac{1}{4}$, $ \frac{1}{4} \times \frac{1}{5}$, ... have a total area of 1.
$\sum\limits_{...
- 21.4k