# Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

421 questions
Filter by
Sorted by
Tagged with
193 views

### Disprove or prove $\sum_{n\ge1} \frac{\vert{\sin n}\vert}{n}$ is divergent. [duplicate]

I want to prove or disprove $\displaystyle\sum_{n\ge1} \frac{{\sin n}}{n}$ is absolutely convergent. I can prove $\displaystyle \sum_{n\ge1} \frac{{\sin n}}{n}$ is convergent by dirichlet's test (Can ...
61 views

### Two problems on real number series

Consider the series: $$a) \sum_{n=1}^\infty \frac{a^n}{ \prod_{k=1}^n \ (1+\sin\frac{1}{2k})}$$ $$b) \sum_{n=1}^\infty \frac{a^n}{ \prod_{k=1}^n \ (1+\tan\frac{3}{2k})}$$ Showing that these two are ...
565 views

260 views

51 views

### Absolute convergence for $\sum_1^n(f(x))^n$ where $f\in C([0,1])$ and $\|f\|_\infty=1$

I am studying for a final exam I have tomorrow and am having trouble solving this practice problem. Let $f \in C [0, 1]$, such that $\|f\|_\infty =1$, where $\| \cdot \|_\infty$ denotes the sup-...
145 views

36 views

### Testing for divergence and convergence [closed]

$$\sum_{k=1}^n (1/2)$$ Why/how does this series diverge? Why doesn't it converge?
24 views

### finding the limit of a series and testing for convergence.

$\lim \limits_{k \to \infty}\cos(kπ) + \sin(k(π/2))$ = undefined, so the series diverges. How is this undefined? isn't undefined when you get 0/0? How would I go about finding out what happens as ...
55 views

95 views

### Absolute convergence of the Fourier series

Given a Fourier series: $$f(x) = \sum_{n=1}^\infty (a_n \sin(nx)+b_n \cos(nx))$$ Show it is absolutely convergent if $$\sum^\infty_{n=1}(|a_n|+|b_n|)$$ is finite. My attempt to show it is as ...
48 views

### Find an example of series which converges only absolutely on $\mathbb Q$
I am currently working on the completeness of metric spaces, so I studied the following theorem: If $E$ is a Banach space then any absolutely convergent series is convergent. Since $\mathbb Q$ is ...
Let $A$ be a normed space where every absolutely convergent series converges in $A$. How do I prove that $A$ is Banach? Let $\sum_{n=1}^\infty x_n$ be an absolutely convergent series in $A$, then ...