# Tagged Questions

For questions on exponential sums, such as $\sum \exp(2\pi ix_n)$.

111 views

### solve equation with sum $\ln x-\ln(\sqrt[2]{ x})+\ln(\sqrt[4]{ x})-\ln(\sqrt[8]{x})+…=2$

How to solve this? Any advice? $$\ln x-\ln(\sqrt[2]{ x})+\ln(\sqrt[4]{ x})-\ln(\sqrt[8]{x})+...=2$$ Next step I do this $\sum\limits_{n=0}^\mathbb{\infty}(-1)^n\ln(x^{\frac{1}{2^n}}) = 2$ But I don't ...
52 views

### sum of fractional powers

Let $z=e^{i\theta}$ for some real number $\theta$, i.e. $z$ is complex number on the unit circle. Is there a formula for $z+z^{1/2}+z^{1/2^2}+\dots+z^{1/2^n}$? Here $n$ is a positive integer. I only ...
30 views

17 views

### Showing that $e^{z}$ series is concentrated around indices close to $|\Re(z)|$

So, I believe (but am having trouble showing) that the power series for $e^{z}$ is concentrated around indices close to $|\Re(z)|$ (or at any rate is negligible beyond those indices). More precisely, ...
37 views

### Help me solve this [closed]

$(1-(1-(1-n)^{-1})^{-1})^{-1}$ Please tell me how to solve the abv sum. The ans is $n$
38 views

### Computation with Exponential Generating Function

I am working a problem on generating functions but am stuck at a technical part which I think is actually supposed to be really easy, but I just don't have much experience with such things. We're ...
34 views

### Is there a way to solve this problem - sum of powers

$N_1^a + N_2^a + \ldots + N_n^a$ Where a is any number. Example $40^7+ 37^7 + 35^7 + 32^7 + ... 1^n=R$ If Iknow what $R$ is and that $N_n$ is restricted to $0 \lt N_n \le 40$, Is there a way to ...
31 views

73 views

275 views

### A “generalized” exponential power series

I'm wondering if $$e^x = \sum_{k=0}^\infty \frac{x^k}{k!}$$ what would this be $$\sum_{k=0}^\infty \frac{x^{k+\alpha}}{\Gamma(k+\alpha)} = \large{?}_{\alpha}(x)$$ for $\alpha \in (0,1)$? ...
187 views

### Standard deviation of phase for a random phasor sum

I have a phasor sum $a e^{j \theta} = \frac{1}{\sqrt{N}} \sum_{k=1}^{N} \alpha_k e^{j \phi_k }$ where $\phi_k = [-\pi, \pi]$, the standard deviation $\sigma_{\phi}$ of the phase is known and the ...
94 views

### quadratic Gauss sum over a power of 2

Is there a general formula for the generalized quadratic Gauss sum defined by $$G(a,b,c)=\frac{1}{c}\sum_{n=0}^{c-1}e\left(\frac{an^2+bn}{c}\right)$$ where $e(x)=\exp(2\pi ix)$ and $c$ is a power of ...
62 views

### Algorithm for smooth exponential curves

I want to plot an exponential curve between 256 and 0. Using the following equasion, I get the resulting data set. (Please note that I am rounding any decimals down to nearest whole number throughout ...
103 views

### How to calculate the sum $x + x^2 +…+ x^n$ [closed]

How can I get the result of this sum: $$x + x^2 +...+ x^n$$
37 views

38 views

### Simple expression:$a - a^{-1}$ = …

I got stuck with one simple expression, I hope get some help with it: If $a-\frac{1}{a}=\frac{3\sqrt7}{7}$, so $a^4+\frac{1}{a^4}=$
209 views

### Sum of exponential square series

I have a infinite sum which I wonder if it will converge to a simpler function $f(r) = \Sigma_n r^{n^2} , r<1$, I also interested in case $r$ is a complex number on unity circle $r = e^{j\omega}$...
How does one use the inverse Mellin transform to prove that the following identity holds? $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n(e^{n\pi} + 1)} = \frac{1}{8}(\pi - 5\log(2))$$ The identity follows ...
### An upper bound for an average exponentiated weighted sum of a vector from $\{-1,1\}^n$
Suppose $\mathcal{S}=\{\mathbf{x}:\mathbf{x}\in\{-1,1\}^n\}$, that is, $\mathcal{S}$ contains all $2^n$ vectors of length $n$ containing -1 and 1. I am interested in the following average: A_{f(n)}...