# Tagged Questions

For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

43 views

### On the convergence of a series $\sum_{n=1}^\infty \left( \frac { p(p+1) \cdots (p+n-1) }{ q(q+1) \cdots (q+n-1)} \right)^n$

I am struggling with the series $$\sum_{n=1}^\infty \left( \frac { p(p+1) \cdots (p+n-1) }{ q(q+1) \cdots (q+n-1)} \right)^n,$$ where $p,q>0$. I have checked the Dirichlet and ratio test so far, ...
15 views

79 views

### Problem concerning the sequence $s_n = 1 + 1/2 +\cdots+ 1/n - \log n$

The question is : Prove that the sequence $\{s_n\}$ where $s_n = 1 + 1/2 +\cdots+ 1/n - \log n$ is convergent. Hence find $\lim_{n \to \infty} \left(1 - 1/2 + 1/3 - ... - 1/2n\right)$. I have ...
58 views

### Sum of series $1−ω^2+ω^4−ω^6+ω^8−ω^{10}+ω^{12}+⋯+ω^{600}−ω^{602}+ω^{604}$

I need to find sum of the series involving cube roots of unity $1−ω^2+ω^4−ω^6+ω^8−ω^{10}+ω^{12}+⋯+ω^{600}−ω^{602}+ω^{604}$. Found it in an old test paper. I applied Geometric Progression Sum Formula....
49 views

17 views

### Alternating series test and divergence test similair?

The alternating series test requires: - Bn to be decreasing - lim Bn (to infinity) to be 0 In my book I see examples where the series fit the first one, but does not fit the second one and then they ...
40 views

108 views

### A closed form of the series $\sum_{n=1}^{\infty} q^n \sin(n\alpha)$

I am having problems with the following series: $$\sum_{n=1}^{\infty} q^n \sin(n\alpha), \quad|q| < 1.$$ No restrictions on $\alpha$. I need to find out whether it converges and if yes, ...
15 views

### What's about $\sum_{n=1}^\infty\frac{\mu(n)}{n}f(X^{\frac{1}{n}})$, where $\mu(n)$ is the Möbius function?

Let $X=\sigma+it$ the complex variable, and $\mu(n)$ the Möbius function. Inspired in Riemann function $R(X)$ I would like to ask you Question. What conditions are required to be satisfied by a ...
19 views

### Construct a function pertaining to the OEIS sequence A131229 (Numbers congruent to {1,7} mod 10)

OEIS sequence A131229 ("Numbers congruent to {1,7} mod 10") begins $\{1, 7, 11, 17, 21, 27, 31, 37, 41, 47, 51,...\}$. I want a function $f(x)$, specifically such that $f(\frac{1}{2}) =\frac{7}{2}$, ...
47 views

### Show $f(x) = \sum_{n=1}^\infty \frac{\sin{nx}}{n^{5/2}}$ is $C^1$

The task is to show $$f(x) = \sum_{n=1}^\infty \frac{\sin{nx}}{n^{5/2}}$$ is a continuous function on $\mathbb{R}$ with a continuous derivative. Showing that the series converges at each point is ...
39 views

### Condition on $(a_n)_{n \in \mathbf{N}}$ for Dirichlet-series $\sum_{n = 1}^\infty \frac{a_n}{n^s}$ to converge for $Re(s) > 1$.

I'm curious about the conditions on a sequence $(a_n)_{n \in \mathbf{N}}$ of real numbers such that the Dirichlet-series $\sum_{n = 1}^\infty \frac{a_n}{n^s}$ converges absolutely for $Re(s) > 1$. ...
21 views

### simplification of functional sum

I have the following infinite sum which seems to be familiar so I was wondering if it admits a possible simplification ! $$\sum_{k\geq 0}\left( g(x_k,x_{k+1})-g(x_k,x_k)\right)$$ where $g$ is only ...
29 views

41 views

### Multiplication of polynomials of the same degree

Consider polynomials of the form $$p(x)=x^{n-2r}\sum_{i=0}^ra_ix^{2i},$$ where \begin{align} r&=n/2, \quad n \quad \text{even},\\ r&=(n-1)/2, \quad n \quad \text{...
70 views

114 views

### How do we prove that $4(3\sqrt2-4)=\prod_{n=1}^{\infty}\left({e^{2\pi(2n-1)}-1\over e^{2\pi(2n-1)}+1}\right)^8?$

How do we prove that $$4(3\sqrt2-4)=\prod_{n=1}^{\infty}\left({e^{2\pi(2n-1)}-1\over e^{2\pi(2n-1)}+1}\right)^8\tag1$$ Rewrite as, to keep it simple Let $a=e^{2\pi(2n-1)}$ 4(3\sqrt2-4)=\...
44 views

### Positive function(or series) that tends to 0 at infinity

Consider these two questions: -If we have an ultimately positive function that tends to 0 at infinity, does this imply that the function has to be decreasing? -Similar questions but for series: if ...
### Convergence of sequence $a_0 := b$ and $a_{n+1} = 2^{a_n}$ in $\hat{\mathbb{Z}}$.
I have a question about the convergence properties of a sequence in $\hat{\mathbb{Z}}$, the completion of $\mathbb{Z}$. It is part of an exercise is due to this syllabus. I got confused somewhere. ...