All Questions
Tagged with convergence or convergence-divergence
22,521 questions
-2
votes
0
answers
38
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Divergence Test [closed]
The divergence test is inconclusive for $f(x) = 1/x$. The sum of the $1/n$'s is in fact divergent, $n\in \mathbb{N}$ and $n$ in $[1, ∞)$. We can say the same for the function $g(x) = 1/x^p$, with $0 &...
- 9
6
votes
1
answer
175
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Is $|\sin(n) | \le 1/n^2$ for infinitely many natural numbers $n$?
I have already finished my analysis classes with good grades. But lately I am playing with Geogebra and wonder if there exists a sequence of natural numbers $(n_k)$ such that $$|\sin{n_k}|\leq\frac{1}{...
- 463
1
vote
1
answer
100
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How to show that the ratio test for $\sum_{n=1}^\infty\frac{(-1)^n\sqrt{n^2+1}}{n\ln n}$ is unity?
According to Wolfram Alpha the ratio test equals $1$ for the series, $\sum_{n=1}^\infty\frac{(-1)^n\sqrt{n^2+1}}{n\ln n}$, hence convergence/divergence is inconclusive. The purpose of this question is ...
- 433
2
votes
6
answers
358
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Find the limit $\lim\limits_{n\to \infty}\sum\limits_{k=1}^{n}\left(\sqrt{1+\frac{k}{n^2}}-1\right)$
I have this limit
$$\lim_{n\to \infty} \sum_{k=1}^{n} \left(\sqrt{1+\frac{k}{n^2}}-1\right)$$ and and I tried to evaluate it in the following way:
First I set $x=\frac{k}{n^2}$ to make the expression ...
- 359
40
votes
7
answers
2k
views
How far can an infinite number of unit length planks bridge a right-angled gap?
Problem: You are standing at the origin of an infinite flat earth. One quadrant of the earth is gone, leaving an infinite abyss. You have an infinite supply of unit-length zero-width planks. How far ...
0
votes
1
answer
81
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Convergence of $\sum_{n=1}^{\infty} a_n,$ $\text{ where }$ $a_n = \prod_{k=1}^{n} \sin^2(2^k x)$ $\text{ and }$ $x \in (-\infty, +\infty)$ [closed]
To determine the convergence of the series $$\sum_{n=1}^{\infty} a_n, \text{ where } a_n = \sin^2 x \sin^2 2x \dots \sin^2 2^{n}x \text{ and } x \in (-\infty, +\infty).$$
I attempted to use the ratio ...
-1
votes
0
answers
38
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Why do we need \rho (utilization)<1 in queuing theory?
I’m studying basic queueing theory, in particular a single–server queue with one arrival stream and one server (a G/G/1 type setup).
Let
A be the interarrival time with mean 𝔼(A),
B be the service ...
- 1
3
votes
2
answers
112
views
Martingales convergence a.s.
Suppose we know that
$P(X=1)=p, P(X=-1)=q=1-p$
We have a martingales
a) $M_n = (\frac{q}{p})^{S_n}$,
b) $M_n = S_n - n(p-q)$
where $S_n =\sum_{i=0}^n X_i$,( $X_i$ iid). How to show (in a simple way) ...
- 51
-3
votes
0
answers
49
views
Divergence of a series based on the sequence [closed]
If a sequence [an] does not converge, that is the limit of "an" as n tends to infinity" does not exist, will the series be referred to as convergent or divergent??
6
votes
1
answer
121
views
Distribution of sum of product [closed]
Let $x_n\overset{p}{\to}c$ and $x_n\overset{d}{\to}N(0,\sigma^2)$ denote convergence in probability to a constant $c$ and convergence in distribution to a random normal variable (with some abuse of ...
- 69
0
votes
2
answers
150
views
Determine the convergence or divergence of $\sum_{n=1}^\infty\frac{1}{n^{\arctan n}}$
There was a problem in my textbook:
Determine the convergence or divergence of $$
\sum_{n=1}^\infty\frac{1}{n^{\arctan n}}
$$
The method my instructor taught me is that, notice that this sum is ...
- 143
7
votes
2
answers
419
views
Series that is known to converge/diverge but for which all these standard tests are inconclusive .
I have noticed that nearly every series I have been asked to analyze its convergence or divergence can be handled by the usual collection of tests: the limit test, Cauchy condensation, the integral ...
- 9,329
1
vote
1
answer
95
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If for every sequence $(x_n)\subset X-\{a\}$ with $\lim_{n\to\infty}x_n=a$, the sequence $(f(x_n))$ is convergent, then $\lim_{x\to a}f(x)$ exists?
I am studying the topic of limits of functions in $\mathbb{R}$ and I want to prove the following
Let $f:X\to \mathbb{R}$ with $X\subset \mathbb{R}$ and $a\in X'$. If for every sequence of points $x_n\...
- 125
0
votes
1
answer
64
views
Is "the nth logarithm test" for series' convergence reliable?
Is it possible to determine the convergence of a series $\sum a_n$ by evaluating $$\lim_{n \to \infty} \log_n(a_n)$$ and comparing the result to $-1$? Specifically, if this limit is less than $-1$, ...
- 58
2
votes
0
answers
92
views
Long term behaviour of the sequence $a_{n+1}=a_0^{a_n}$ for $a_0\in\mathbb{C}$
I am interested in the long term behaviour of the sequence $a_{n+1}=a_0^{a_n}$ where $a_0\in\mathbb{C}$. I am aware there are many sources discussing convergence of this sequence(for example) but I am ...
- 1,313