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

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0
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0answers
39 views

Convergence in probability of the inverse of a sequence of random variables.

Let $X_n$ be a sequence of random variables such that $X_n>0$ a.s. for all $n$ and such that $X_n\stackrel{p}{\rightarrow} X$ with $X>0$ a.s.. I want to show that ...
1
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0answers
17 views

Investigating the cardinality of the set of all subsequences of any arbitrary sequence?

This is a non-precisely formulated question recently come to mind: How to investigate the cardinality of the set of all subsequences of an arbitrarily given sequence? Or can we possibly determine the ...
2
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4answers
60 views

Prove that $\lim_{n\to\infty}b_n = a$

Let $\lim_{n\to\infty}a_n = a$. Let $(b_n)$ be a sequence satisfying $b_{n+k} = a_{n+l}$ for some $k,l\in\mathbb N$ and all $n\in\mathbb N$. Prove that $\lim_{n\to\infty}b_n = a$ ...Well, first, I ...
3
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2answers
101 views

A limit problem about $a_{n+1}=a_n+\frac{n}{a_n}$

Let $a_{n+1}=a_n+\frac{n}{a_n}$ and $a_1>0$. Prove $\lim\limits_{n\to \infty} n(a_n-n)$ exists. In my view, maybe we can use $${a_{n + 1}} = {a_n} + \frac{n}{{{a_n}}} \Rightarrow {a_{n + 1}} - ...
2
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2answers
37 views

Choose for what constants the series converges or diverges [closed]

I have a series $$ \sum_{n=1}^\infty \frac{1}{a^n + 1} $$ where $a > 0$. I need to find for which values $a$ the series is convergent and divergent. I have no idea what to do. I don't think I ...
10
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2answers
250 views

How to prove this recursive sequence converges to $\sqrt 2$?

Let $a_0,a_1>0$ be given. Consider the recursive sequence $$a_{n+2}=\frac{1}{a_{n+1}}+\frac{1}{a_n}$$ Prove that $a_n\to\sqrt2$. I attempted to find a bound for $a_n$ but so far I have only got ...
1
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1answer
38 views

Group and a sequence of order of elements

Give an example of a group $G$ such that the sequence $s_n= \dfrac{o(g_1) +o(g_2)+...+o(g_n)}{n}$ ($o(g_n)$ is the order of element $g_n$) has $\lim_{n\to\infty} s_n \in \mathbb Q^c$ ($\mathbb Q^c ...
0
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1answer
26 views

Give a sequence such that: $\forall \epsilon >0 $ $\exists N \in \Bbb N$ such that if $N \le n \le N+2$ then $|X_n - L| \ge \epsilon $

I have to solve a lot of excercises like these but I am not sure of my answers and in some cases I do not what to do. Give a sequence $X_n$ and a number L such that: 1) $\forall \epsilon >0 $ ...
2
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2answers
33 views

Prove that if lim $a_n=L$ then lim $p_n=L$, with $p_n= \frac1n \sum_{k=1}^n a_k$.

I have to do this: Prove that if lim $a_n=L$ then lim $p_n=L$, with $p_n= \frac1n \sum_{k=1}^n a_k$. And show that the converse is false. What I have done is: a)Due to $a_n=L$ exists $N$ such that ...
0
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1answer
20 views

Creating a chain rule from a series of similar expressions

I'm doing some calculations on trial and error basis although I'm using the same formula in each step: c1 = c0 - c0*R + F c2 = c1 - c1*R + F c3 = c2 - c2*R + F ...
0
votes
1answer
54 views

Calculating the periodicity of a rose curve

I'm having fun generating pretty pictures using math, and have started to draw rose curves. Thing is, I'm not sure when to stop drawing. The formula is the same as from the wikipedia entry - a=angle, ...
1
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0answers
30 views

Prove: $\sum_{n=1}^{\infty} \frac1{n^{3/2}}$ converges

I have to prove the convergence of the first series: 1)$\sum_{n=1}^{\infty} \frac1{n^{3}}$ 2) $\sum_{n=1}^{\infty} \frac1{n^{1/2}}$ 3)$\sum_{n=1}^{\infty} \frac1{n^{3/2}}$ I have proved the ...
1
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1answer
51 views

Exponential generating series of binomial coefficients: $\sum_{k=0}^\infty{ k \choose j}\frac{x^k}{k!} $

I'm wondering if anyone knows what it is? The exponential generating series I have in mind is $$ f_j(x) = \sum_{k=0}^\infty { k \choose j} \dfrac{x^k}{k!}. $$ Thanks!
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2answers
47 views

Nth-term test for $\sum_{n=0}^{\infty} \frac{n-\sqrt{n}}{n^2 \sqrt{n}} $?

$\sum_{n=0}^{\infty} \frac{n-\sqrt{n}}{n^2 \sqrt{n}} $ supposes to diverge. But when the nth-term test is applied to it, like the way it is done here, the limit will approach 0 when n goes to ...
1
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2answers
39 views

Binary Sequence of Single Bit Transitions

First of all, I'll have to say that I believe this problem has no solution, but I'm unable to prove it. Here is the problem: I need an algorithm to generate a sequence of all possible transitions of ...
0
votes
1answer
34 views

Is a positive series convergent if the terms decrease?

Suppose positive real numbers $n_1>n_2>n_3>n_4...$ with these properties are given and you have the sum of $ n_1+n_2+n_3+n_4...$ Is it possible to determine on the basis of this information ...
1
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0answers
23 views

convergence of a sequence satisfying functional inequality

I am stuck with this "funny" problem. Let $(u_n)_{n \in \mathbb{N}}$ be a sequel of real numbers such that $$ \forall k \in \mathbb{N}^*, \quad \exists C_k \in \mathbb{R}^+, \quad \forall n \in ...
1
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1answer
27 views

a variant of Basel problem (including a real number shift)

is a solution of this infinite series known? $\sum_{n=1}^\infty \frac{1}{(n-f)^2}$ where $f$ is a generic real number (but cannot be integer) Thank you for your help
0
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1answer
81 views

Positive Sequence, convergent series, is the sequence decreasing?

If there is a positive sequence and the series of this sequence converges from n to infinity, must the sequence be decreasing? bounded or have a decreasing sequence? I'm not sure how to prove or ...
1
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2answers
72 views

Prove that $\lim_{n\to\infty} |a_n| = |a|$

Let $\lim_{n\to\infty} a_n = a$. Prove that $\lim_{n\to\infty} |a_n| = |a|$. First of all, I wonder about what does it mean the statement. I am just studying the theory today. I actually know that ...
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2answers
469 views

Does the series 0.5, 0.4, 0.3, 0.2, 0.1, 0.09, 0.08… diverge to infinity?

If one has a sequence 0.5+0.4+0.3+0.2+0.1+0.09+0.08+0.07+0.06+0.05... and this series continues to infinity, does the sum of this series reach infinity? If so why? If not why not? So far I have been ...
4
votes
1answer
65 views

Very tricky radius of convergence involving a non-monotonic summand,

Edit: hints or solutions are welcome. This question is something weird that I have not seen before, so I don't have much of a starting point - hadamard's radius of convergence formula also doesn't ...
0
votes
2answers
17 views

Unspecific limits in a summation sign

I'm working on a probability problem, displayed in image 1, in which I encounter a problem. The solution is on the left and the original assignment on the right. The problem is that I don't get the ...
2
votes
2answers
225 views

Uniform convergence on closed interval

If $(f_n)$ converges pointwise on $[-1,1]$ to $f$, and converges uniformly on $[-r,r]$ for every $r \in (0,1)$, does it follow that the convergence is uniform on $[-1,1]$? Also $f$ is continuous and ...
1
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0answers
28 views

Conditions on a permutation of $\mathbf{N}$ such that rearrangements of series don't change convergence

I'm doing exercises from Dieudonné's Foundations of modern analysis. Problems 5.2.2 and 5.2.3 are concerned with conditions under which rearrangements of series converge. Let $E$ be a normed space ...
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0answers
27 views

Determining convergence or divergence of a real sequence.

I saw the following question in a competitive exam : I am supposed to determine whether the sequence of real numbers given by $\cos(\frac{1}{2} \arctan((-\frac{n}{2})^n)))$, is monotone or not and ...
0
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1answer
31 views

If $\sum a_m$ exist then $\sum a_{2k}$ need not exist.

If $\sum a_m$ exist then $\sum a_{2k}$ need not exist I am unable to find a counter example of the above fact. Is it possible?
0
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1answer
82 views

A comprehensive problem about $\sum {\frac{{n\cosh \left( {nx} \right)}}{{\sinh \left( {n\pi } \right)}}} $

Prolem: ${\mathop {\lim }\limits_{x \to {\pi ^ - }} \mathop {\lim }\limits_{n \to \infty } \left( {{n^2}\left( {1 - \sin \frac{\pi }{{2n}} \cdot \sum\limits_{k = 1}^{n - 1} {\sin \left( {\frac{{\left( ...
2
votes
2answers
314 views

Limits at infinity involving integrals

I'm asked to find $$\lim_{n\to\infty}\int_{n}^{n+1} e^{-x^2}dx $$ Trouble is, I have no real idea how to go about evaluating that integral -- u-substitution doesn't really seem to work, at least. I ...
1
vote
1answer
16 views

cardinality of a countable sequence of powersets

Given a set $S_1$ of cardinality $\kappa$, we can construct the sequence $\langle S_1, S_2, S_3 ... \rangle$, where $S_i = \wp(S_{i-1})$, for all $i > 1$. If $S$ is finite, so that $\kappa < ...
0
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0answers
19 views

Proofs of boundedness

I'd like to prove that $(-1)^n$ is bounded. Intuitively I know that it is, because the only values it assumes are -1 and 1, but it doesn't converge so I can't say that it's bounded because it ...
1
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3answers
81 views

Sum of the geometric series $ \sum_{n=0}^{\infty} \, (-1)^{n}(x+6)^{n}$

Consider the geometric series $\; \displaystyle \sum_{n=0}^{\infty} \, (-1)^{n}(x+6)^{n} .$ The open interval of convergence is ? Find the sum of the series on this interval ? Formula for ...
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1answer
49 views

Can you find a convergent sequence that is not bounded? [closed]

Can you find a convergent sequence that is not bounded? Examples please
0
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0answers
26 views

suppose that $E\subset\mathbb{R}$ is a nonempty bounded set, and that $\sup E\notin E$.

Prove that there exists a strictly increasing sequence $\{x_n\}$ that converges to $\sup E$ such that $x_n\in E$ for all $n\in\mathbb{N}$ My attempt at a solution: By the Approximation Property, ...
2
votes
3answers
38 views

Convergence of the series of terms $\frac{(\log(i))^k}{i^2}$ using the comparison test

I was wondering if someone knows a comparison test argument for the convergence of $$\sum_{i=2}^{\infty} \frac{(\log(i))^k}{i^2}$$ for all $k \gt 0$. If possible, I am particularly interested in a ...
2
votes
2answers
86 views

Does the series $1+\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\frac{1}{7}+\frac{1}{8}-\frac{1}{9}+\cdots $ converge or diverge?

Does the series $1+\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\frac{1}{7}+\frac{1}{8}-\frac{1}{9}+\cdots $ converge or diverge?
3
votes
0answers
44 views

Is $\sum_n\exp(ian+ibn^2+icn^3)$ known in terms of anything else?

For arbitrary $a,b,c$, does the series $$F(a,b,c)=\sum_{n=-\infty}^\infty\exp\left(ian+ibn^2+icn^3\right),$$ i.e. an evenly-weighed series of exponentials of cubic polynomials, converge to anything ...
3
votes
2answers
37 views

Summate using Abel method series $ \sum_{n=1}^{\infty} \sin{\theta n} $

Summate using Abel method series $ \sum_{n=1}^{\infty} \sin{\theta n} $. For Abel sum we should consider functional series $ \sum_{n=1}^{\infty} \sin{\theta n} \cdot x^{n-1}$, find the sum $ S(x) $ ...
7
votes
2answers
207 views

upper bound of a double sequence

Let $\{ a_{k,m} \}$ be a doublely indexed sequence of positive numbers satisfying: $a_{1,n}\leq \frac{1}{n+1}\quad $ and $\quad a_{k,m} \leq \frac{1}{m+1}(a_{k-1,m+1}+L a_{k-1,m+2})\quad \quad (1)$ ...
0
votes
1answer
23 views

expressing $p , p(p+1) , p(p+1)(p+2)$ as a series

I'm working on arithmetical analysis and more specifically on finite differences. I want to create a series consisting of the following terms : $$f(x_{0} + ph) = f_{0}+ p\Delta f_{0}+ ...
0
votes
1answer
47 views

Show that $N\sum_{n = 2}^{\infty} \frac{t^n}{nN^n}$ converges uniformly to $0$

I want to show that $$N\sum_{n = 2}^{\infty} \frac{t^n}{nN^n}$$ converges uniformly to $0$ as $N \to \infty$ on a bounded interval. I have done a very few problems on showing that the power series ...
0
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0answers
6 views

Globally convergent series for Dirichlet L-Functions

I've seen the two globally convergent series representation for the classic Zeta Function on the wiki page. Do globally convergent series exist for all Dirichlet L-Functions? I couldn't find any ...
0
votes
2answers
84 views

Prove that if a sequence converges then $\lim_{k \to \infty} kb_k = 0$ [duplicate]

We have that $(b_n)$ is a sequence of decreasing, non-negative real terms. We wish to show that if $\displaystyle \sum_{i=1}^{\infty} b_n$ converges then it must be the case that $$\lim_{k \to \infty} ...
-1
votes
3answers
48 views

What is the limit of this sequence for $n\to\infty$? [closed]

How can I calculate the limit of the following sequence? $$a_{n}=\sqrt[n]{4+\frac{n-1}{n+1}}$$
0
votes
1answer
39 views

Limit of: $a_{n+2} =\frac{2}{a_n + a_{n+1}}$ [duplicate]

$\{a_n\}$ is a real sequence, $a_1 > 0 $, $a_2 > 0$ and for all $n>2$ : $$a_{n+2} =\frac{2}{a_n + a_{n+1}}.$$ Prove that: $\lim_{n\to \infty} a_n = 1$.
1
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0answers
32 views

Is there a citation for this interesting relationship?

While preparing a manuscript for publication, I recently found the following relationship: 1/(1+ratio) + 1/(1+(1/ratio)) = 1 which appears to hold true for any ratio or value except 0 and -1 I ...
1
vote
0answers
25 views

Summation of an infinite q series

When calculating a Partition function, I encounter the following summation $$\sum_{n=0}^{\infty} x^n q^{n^2}.$$ I know that the sum$\sum_{n=-\infty}^{\infty} x^n q^{n^2}$ is a Theta function , but I ...
1
vote
2answers
24 views

Number of binary strings with sub-string constraint

How many binary strings of length $n$: $a_1a_2\dots a_n$ are there, such that for every sub-string of k consecutive numbers $a_ia_{i+1}\dots a_{i+k-1}$ and $\forall k, 1 \leq k \leq n$, the difference ...
2
votes
3answers
101 views

Find the value of $\sum_0^n \binom{n}{k} (-1)^k \frac{1}{k+1}$

Find the value of $\sum_0^n \binom{n}{k} (-1)^k \frac{1}{k+1}$. Writing out several terms, I think the answer is $1/(n+1)$, but I'm struggling to prove this. I would greatly appreciate it if someone ...
2
votes
1answer
55 views

Show that $\displaystyle f_n(x) = \frac{x}{1+n^2x^2}$ converges uniformly to $0$ on $[0, \infty)$

Trying to use the definition of uniform convergence where $|f_n(x) \to f(x)| \to 0$ as $n \to$ ∞ but not sure how to apply this definition to the problem