Tagged Questions
3
votes
3answers
174 views
the sum of a series
I am stuck on the computation of the following sum:
$$\sum_{k=0}^{\infty} {\Big( {\frac{q}{k+1}} \Big)}^k ,$$
where $k$ is a natural number, and $0<q<1$.
7
votes
1answer
226 views
Solving a formal power series equation
I want to find a function $f(x,y)$ which can satisfy the following equation,
$$\prod _{n=1} ^{\infty} \frac{1+x^n}{(1-x^{n/2}y^{n/2})(1-x^{n/2}y^{-n/2})} = \exp \left[ \sum _{n=1} ^\infty ...
1
vote
1answer
56 views
Any dominance between these two functions?
Let $f\left(x\right):=e^{x}+e^{-x}+2$ and $g_{\beta}\left(x\right):=4e^{\beta x^{2}}$.
Do there exist $a>0$ and $\beta>0$, such that $f\left(x\right)\le g\left(x\right)$
for all $x$, $0\le x\le ...
2
votes
4answers
150 views
Formula for calculating $\sum_{n=0}^{m}nr^n$
I want to know the general formula for $\sum_{n=0}^{m}nr^n$ for some constant r and how it is derived.
For example, when r = 2, the formula is given by:
$\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$
...
0
votes
1answer
61 views
Manipulating Indices on Series
I have a series:
$$
\sum_{n_1-l_1=0}^{\infty}\sum_{n_2-l_2=0}^{\infty}\sum_{n_3-l_3=0}^{\infty}a_{n_1-l_1,n_2-l_2,n_3-l_3}r^{n_1-l_1}s^{n_2-l_2}t^{n_3-l_3}
$$
Which is equal to another series:
$$
...
0
votes
1answer
151 views
Sum of the polynomial roots raised to a power. How to prove?
Problem:
If we have a polynomial $f$ with a derivative $f\,'$ and quotient $q$ function defined as:
$$q(x)=\sum_{i=1}^{\infty}a_ix^{-i}=\frac{f\,'(x)}{f(x)},$$
and the roots of $f$ are ...
1
vote
1answer
156 views
How to find the sum? Based on logarithm function expansion
The problem:
How to find the sum?
$$-\sum_{i=1}^{\infty}\frac{(-x)^{i\; \bmod(k-1)}}{i}$$
Details:
I tried find this sum using Mathematica
...
