All Questions
112
questions
2
votes
1
answer
31
views
If a sequence $a_n$ satisfies the following two properties, does $\sum_{k=1}^{\infty} (\sum_{n=1}^{\infty} \frac{1}{a_n^k} - L)$ converge?
Let $a_n$ be a positive, increasing sequence satisfying the following two properties:
$S_k :=\displaystyle\sum_{n=1}^{\infty} \frac{1}{a_n^k}$ converges for all $k \in \mathbb{N}$.
And $\displaystyle\...
- 340
3
votes
1
answer
63
views
Proving Existence of Sequence and Series Satisfying Hardy-Littlewood Convergence Condition Prove that for every $\vartheta, 0 < \vartheta < 1$,
Prove that for every $\vartheta, 0 < \vartheta < 1$, there exists a sequence $\lambda_{n}$ of positive integers and a series $\sum_{n=1}^{\infty} a_{n}$ such that:
\begin{align*}
(i) & \...
6
votes
1
answer
229
views
Is it always true that if $x_n\to0$, $y_n\to0$ there exist $\epsilon_n\in\{-1,1\}$ such that both $\sum\epsilon_nx_n$and $\sum\epsilon_ny_n$ converge? [duplicate]
I saw this interesting problem:
Let $x_n$ and $y_n$ be real sequences with $x_n \to 0$ and $y_n \to 0$ as $n \to \infty$.
Show that there is a sequence $\varepsilon_n $ of signs (i.e. $\varepsilon_n \...
- 6,620
2
votes
1
answer
111
views
For $a>1$ and a fixed $N$ does $\sum\limits_{n=1}^\infty \prod\limits_{k=0}^{n-1} \left(1- \frac{a}{a+N +k} \right)$ converge?
I was trying to find a proof for Raabe's Ratio test: If $x_n$ is a positive sequence of real numbers, if $\lim\limits_{n \to \infty } n \left(\frac{x_n}{x_{n+1}}-1 \right) >1$ then $\sum\limits_{...
- 6,620
2
votes
1
answer
87
views
Convergence of the series $f(a_n)a_n$
Let $f : \mathbb{R} \to \mathbb{R}$ a monoton function in $[−r, r]$ for some $r>0$. Prove that the if $$\sum a_n$$ converge absolutely then $$\sum f(a_n)a_n$$ converges absolutely
I have not idea ...
- 145
1
vote
1
answer
123
views
Convergence of summation of complex exponentials with alternating exponent
Related to my previous question, consider $$f(s)=\sum_{k=1}^{\infty} \exp(-s(-2)^k)$$where $s\in\mathbb{C}$ is a complex number. According to the Willie Wong's comment, $f(s)$ diverges when $\Re\{s\} \...
- 4,359
1
vote
0
answers
50
views
Does this series converge ? $\sum_{n=1}^{\infty} (-1)^n(\sqrt[n]{a}-1), a \geq 1$ Hint: $\lim_{n \to \infty} \frac{\sqrt[n]{a}-1}{1/n} = \ln{a}$
Does this series converge ?
$$\sum_{n=1}^{\infty} (-1)^n(\sqrt[n]{a}-1), a \geq 1$$
Hint: use the fact that
$$\lim_{n \to \infty} \frac{\sqrt[n]{a}-1}{1/n} = \lim_{h \to 0} \frac{a^h-1}{h} = \ln{a}$$
...
- 1,135
1
vote
1
answer
86
views
Find radius of convergence of the power serie $\sum_{n=1}^{\infty} (\sqrt{n} - 1)^{\sqrt{n}}z^n$
Find radius of convergence of the power serie $\sum_{n=1}^{\infty} (\sqrt{n} - 1)^{\sqrt{n}}z^n$
I first tried to use the Root Test.
$L = \lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} \...
- 1,135
0
votes
2
answers
69
views
Find radius of convergence of $\sum_{n=1}^{\infty} \frac{(-1)^n(2z)^n}{n}$
Sorry for my last duplicate question. But for this question here, I did not find the same question
Find radius of convergence of $\sum_{n=1}^{\infty} \frac{(-1)^n(2z)^n}{n}$
$L = \lim_{n \to \infty} |\...
- 1,135
0
votes
0
answers
14
views
Find radius of convergence of $\sum_{n=1}^{\infty}\frac{n!}{n^n}z^n$ [duplicate]
Find radius of convergence of $\sum_{n=1}^{\infty}\frac{n!}{n^n}z^n$
My attempt:
$L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$ with $a_n = n!/n^n$. This gives
$L = \lim_{n \to \infty} |(n+1)\frac{n^...
- 1,135
1
vote
0
answers
56
views
Prove for a sequence ($a_n$)$_n$ with converging partial sum $s_n = \sum_{k=1}^{n}a_k$ it holds that for a bounded, monotonically decreasing sequence
Prove that for a sequence ($a_n$)$_n$
with converging partial sum
$$s_n = \sum_{k=1}^{n}a_k$$ , it holds that for a bounded, monotonically decreasing sequence ($c_n$)$_n$, the series
$$\sum_{n=1}^{\...
- 1,135
3
votes
2
answers
123
views
Does this sum converge? Does it converge absolutely? $\sum_{n=1}^{\infty} \frac{(-1)^n}{n-\sqrt{n-1}}$
Does this sum converge? Does it converge absolutely?
$$\sum_{n=1}^{\infty} \frac{(-1)^n}{n-\sqrt{n-1}}$$
I first checked absolute convergence. Taking the absolute value of the term, we get $\frac{1}{...
- 1,135
5
votes
1
answer
106
views
Does this serie converge ? $\sum_{n=1}^{\infty} \frac{1}{n^3 - 5n}$
Does this serie converge ? $$\sum_{n=1}^{\infty} \frac{1}{n^3 - 5n}$$
We have $$n^3 - 5n < n^3 \implies \frac{1}{n^3-5n} > \frac{1}{n^3}$$
We know $\frac{1}{n^3}$ converges because it's a p ...
- 1,135
1
vote
1
answer
46
views
Change signs of sequence elements such that sum of elements converges
Suppose there is a sequence like $(a_i)_{i \ge 1}$, such that infinite sum of sequence diverges and for every element $|a_i|\le1$ and $a_n \to 0$. Is it possible for any sequence to change sign of ...
1
vote
1
answer
59
views
Cezaro mean convergence implies regular convergence outside of a rare set
I have a bounded sequence of non-negative real numbers ($0 \leq a_i < C$), which Cezaro mean converges to $0$:
$$
\lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^{n - 1} a_i = 0
$$
How do I proof that ...
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