# Tagged Questions

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 $0<q<1$.
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 ...
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 ...
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: $\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$ ...
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:  ...
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 ...
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 ...
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