For questions about recurrence relations, convergence tests, and identifying sequences

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24 views

Does such formula exist $\lim_{n\to \infty}\frac{\log_ex_n}{x_n-1}=1$

Does such a formula of limit related to sequences exist? $$\lim_{n\to \infty}\frac{\log_ex_n}{x_n-1}=1$$ where $x_n$ is a sequence .
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0answers
25 views

Does $\lim \frac {a_n} {b_n}$ exist and $\lim a_n \neq 0$ imply $\lim b_n$ exist?

Suppose $\lim_{n \rightarrow \infty} \frac {a_n} {b_n}$ exist and $(a_n)$ converges to some number $k \neq 0$. Is it then possible to conclude that $(b_n)$ converges ? Also, suppose $\lim_{n ...
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1answer
35 views

Limit Superior Proof: Why Is the Condition $\beta \ne 0$ Necessary?

Suppose that $\{a_n\}, \{b_n\}$ are sequences of nonnegative real numbers with lim $b_n = \beta \ne 0$ and limsup $a_n = \alpha$. Prove that limsup $a_nb_n = \alpha\beta$. The text gave part of the ...
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0answers
23 views

Nested recurrence sequence with interesting properties

This is my first post here, thanks for stopping by. The question as written below comes from the book 'A Concise Introduction To Pure Mathematics'. I've included my working (this isn't a homework ...
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1answer
43 views

Limit of a sequence defined by recursive relation : $ a_n = \sqrt{a_{n-1}a_{n-2}}$

We're given a sequence defined by the recursive relation: $$a_n = \sqrt{a_{n-1}a_{n-2}}$$ $a_1$ and $a_2$ are positive constants. We have to show the following: The sequences $\{ b_n \} = \{ ...
3
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2answers
36 views

Is it true that: $\sum _{i=0}^\infty\left( \prod _{j=0}^{i-1}p_{j}-\prod _{j=0}^{i}p_{j} \right)=\left(1-\prod _{j=0}^{\infty }p_{j}\right)$?

Okay, I'm trying to prove some exercise, and after hours and hours of trying, I think I can show it, if the following is true: $$\sum _{i=0}^\infty\left( \prod _{j=0}^{i-1}p_{j}-\prod _{j=0}^{i}p_{j} ...
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1answer
30 views

Finiteness of the Supremum of Inner Product of Two Finite Sum Positive Sequence

Let $$A = \Big\{(a_1,a_2,\dots)\ \Big|\ a_i\ge 0, \sum_{i=1}^\infty a_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum_{i=1}^\infty a_ib_i\ \bigg|\ (a_i)_{i=1}^\infty,\, (b_i)_{i=1}^\infty \in ...
1
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1answer
35 views

Problem with itinerary of a coding problem with infinite 1's

If $f(x)=2x \ mod \ 1$ on $[0,1)$. Then if we code $x \in [0,1)$ with its itinerary w.r.t. the partition $P_0=[0,1/2)$ and $P_1=[1/2,1)$. Can you show that there is no point $x$ whose itinerary has ...
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2answers
40 views

Proving a sequence is convergent

Let $\ (x_n )_{n \ge 0} $ be a convergent sequence . Prove that another sequence $\ (y_n )_{n \ge 0} \ $ defined as $ x_n = y_n + 2y_{n + 1} $ is convergent as well . I tried mixing the $\ ...
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1answer
23 views

Multiple representations of ternary expansions of numbers

$x \in [0,1]$. If in binary expansions ie series $\displaystyle x = \sum_{i=1}^{\infty} \frac{x_i}{2^i}$ where each $x_i \in \{0,1\}$ we identify the sequences $\underline{x}$ and $\underline{x}'$ ...
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0answers
23 views

Is lim sup $s_{n}$ larger than all other tails of $s_{n}$?

I was reading a proof and it said that let $q$ = lim sup $s_{n}$. Then q is the largest possible value any $s_{n}$ in a tail of the sequence of $s_{n}$ can attain. So, sup $T_{m} \leq q$ where ...
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2answers
197 views

Limit of $\int_0^1\frac{B_{2n+1}\left(\left\{\frac1x\right\}\right)}{x}dx$

Set $$u_n= \int_0^1 \frac{B_{2n+1}\left(\left\{\frac1x\right\}\right)}{x}dx\tag1$$ where $\left\{t\right\}=t-\lfloor t \rfloor$ denotes the fractional part of $t$ and where $B_n(\cdot)$ are the ...
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1answer
24 views

Find the sum of n terms of a series

Find the sum of n terms of series whose $n$th term is $\frac{n^2(n^2-1)}{4n^2-1}$.
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0answers
28 views

Enumerating the rationals in $[-1,1]$ so that the average converges to a prescribed limit $t\in [-1,1]$

Suppose $(q_n)$ is an enumeration of the rationals in $[-1,1]$ (meaning $q:\mathbb{N}\rightarrow \mathbb{Q}\cap [-1,1]$ is a surjection) and let $t\in [-1,1]$. Show that there is a reordering ...
1
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1answer
20 views

Sequence increase/decrease

I need help determining if the following sequences are increasing, decreasing or not monotnic. Any help would be great. I think that for c) it is not monotonic as well as for a. a) ...
2
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3answers
62 views

Does $\sum_{n=1}^\infty \frac{(-1)^n}{n^{1+\frac{1}{n}}}$ converge?

I want to use the alternating series test here, but I've just been told that it won't work because it's not monotonically decreasing. However, if the alternating harmonic series converges then don't ...
2
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1answer
42 views

Uniform convergence

How would you show that $$f(x)=\sum_{n=2}^\infty \frac{\sin(2\pi n x)}{n\log n}$$ converges uniformly on $x\in[0,1]$. The pointwise convergence can be proved by Drichlet test
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23 views

Showing that the topologist's sine curve is not path connected using an argument of sequences.

I am familiar with many proofs of the fact that the set (defined below) is not path connected. My favorite uses the fact that $[0,1]$ is compact and another good one uses the intermediate value ...
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1answer
13 views

Help with alternating series

I'm trying to get an explicit formula for the series $\sum_{n=1}^\infty\frac{(-1)^n}{4n+3}$ What I tried to do was use the taylor series for the natural logarithm: $-\ln(1-z)=\sum_{n=1}^\infty ...
0
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1answer
47 views

Expression for $\sum_n n^a/n!$ [duplicate]

I'm wondering if there is a general solution for $$S_a =\sum_{n=0}^{\infty} \frac{n^a}{n!}$$ with $a \in \mathbb{Z}$ and $a > 0$. From Mathematica: $$S_1 = 1e$$ $$S_2 = 2e$$ $$S_3 = 5e$$ $$S_4 = ...
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2answers
95 views

Show that $\,a_n=f(1)+f(2)+\cdots+f(n)-\int_1^n f(x)\,dx\,\,$ converges

Let $\,f:[1,\infty)\to \mathbb R\,$ be a decreasing and lower bounded function. Show that the sequence $\{a_{n}\}_{n\in\mathbb N}$ defined as: $$ a_n=f(1)+f(2)+\cdots+f(n)-\!\int_1^n\!\! f(x)\,dx, $$ ...
1
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1answer
34 views

How to obtain this geometric progression

How do I obtain this from the formula of the geometric progression (which I 'only' know as $1+q+q^2+...+q^{n-1} = \frac{1-q^n}{1-q}$)? $$\frac{x_1^p-x^p}{x_1^q-x^q} = ...
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2answers
44 views

If $\frac{[x_n]^n [1-[x_n]^n]}{(1-x_n) n} = a$, is $[x_n]^n$ increasing for $n\geq 3$?

Let $x_n$ be the solution to $\frac{x^n [1-x^n]}{(1-x) n} = a$, where $x \in [0,1], a \in [0,1]$ and $n \in \mathbf{N}$. I want to prove that $[x_n]^n$ is increasing in $n$ for $n\geq 3$. (From ...
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1answer
70 views

Evaluating Telescopic Sum $ \sum\frac{n}{1+n^2+n^4} $

How to evaluate following $$ \sum_{n=1}^{\infty}\frac{n}{1+n^2+n^4}$$ I posted my way as an answer, Is there another Interesting approach to evaluate this sum of series?
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1answer
20 views

Using the comparison test to determine series' convergence

I'm having trouble figuring out how to use the comparison test to check if an infinite series converges or diverges. I put two problems that I have to solve, does anyone have any input on this? The ...
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1answer
18 views

Bounded and absolute convergent real sequences

I have problems with this question: Denote $E$ the set of all real sequences $\{a_n\}$ such that $|a_n|\le 1$ for every positive integers $n$, $l^1$ be the set of all real sequences $\{a_n\}$ such ...
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0answers
25 views

Convergence of series, cesaro summability

Suppose that series $\sum_{n=1}^\infty na_n^2<\infty$ and $\sum_{n=1}^\infty a_n$ is Cesaro summable. How do you show that $\sum_{n=1}^\infty a_n$ converges? I know that if $\lim_{n\rightarrow ...
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0answers
28 views

Does the series $\sum_{n=1}^\infty \frac{(-1)^n}{n^{1+\frac{1}{n}}}$ converge?

Attempt: We can write the terms in the series as $(-1)^n a_n$ where $$ a_n = \frac{1}{n^{1+\frac{1}{n}}}< \frac{1}{n}.$$ And since $\lim_{n \to \infty} \frac{1}{n} = 0$ and is monotonically ...
1
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1answer
45 views

Comparing fractals

Is there a way to compare if two fractals are "isomorphic"? I'll give an example of what I mean. Consider the following two fractals. First we have the Sierpinski triangle, and next we have the ...
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2answers
36 views

Proving part of the theorem that a continuous function on the bounded interval is uniformly continuous

I am asked to prove the following. Suppose $D$ is a closed and bounded subset of $R$ and suppose $f:D\to R$ is continuous on $D$. Then $f$ is uniformly continuous: $ $Proof: Suppose by contradiction ...
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1answer
44 views

Convergence of Difference of Sequences

Suppose $\{x_n\}$ and $\{y_n\}$ are sequences in $\mathbb{R}$ such that $$\lim_{n \to \infty} x_n = \lim_{n \to \infty} y_n = \infty$$ and $$\lim_{n \to \infty} \frac{x_n}{y_n} = q.$$ What can we say ...
2
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2answers
54 views

Prove sequence $S_n$ converges

If $S_1 = \sqrt{2}$, and $S_{n+1} = \sqrt{2 + \sqrt{S_n}}$ (n = 1,2,3....), prove that $\{S_n\}$ converges, and that $S_n < 2$ for all $n \in \Bbb{N}$ This is one the questions ...
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1answer
36 views

Show that there is a series in R^infinity has some term greater than or equal to 1/n but that also is arbitrary close to the zero sequence.

Consider $\mathbb{R}^\infty = \{(a_n): \sum_{n = 1}^{\infty} a_n^2 < \infty\}$ with the metric $d((a_n$), ($b_n$)) = $[\sum_{n=1}^{\infty} (a_n - b_n)^2]^{1/2}$. Let $A = \{(a_n) : |a_n| < 1/n ...
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1answer
18 views

Fourier cosine series for a interval $[0, l]$

It is asked to find the Fourier Cosine Series for the function defined by $$f(x) = \cos \frac{\pi x}{l}, x \in [0, l/2]$$ $$f(x) = 0, (l/2, l]$$ I thought it should be $$\frac{a_o}{2} + \sum a_n ...
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2answers
58 views

Find the sum $\sum_{n=1}^\infty \frac{n}{(1+x)^{2n+1}}$

Find the sum $$\sum_{n=1}^\infty \frac{n}{(1+x)^{2n+1}}.$$ Indicating the interval of convergence for $x$. My attempt: Let $ t=\frac{1}{x+1}$. Then, applying the root test, $$\lim_{n\to \infty} ...
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5answers
77 views

Does the series $\sum_{n=1}^\infty \frac{n+1}{n^3+10n}$ converge?

Using the ratio test, we evaluate: $$\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty}\left| \frac{(n+1) + 1}{(n+1)^3 + 10(n+1)} \cdot \frac{n^3+10n}{n+1} \right| = \lim_{n ...
0
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3answers
41 views

Prove sequence $\frac{5n+1}{n^{5}-2}$ is convergent

Please can someone help prove that the sequence $\frac{5n+1}{n^{5}-2}$ is convergent from first principles? Thanks
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2answers
68 views

Summation of convergent series

Sum the series $$\sum_{n=0}^{\infty}\left(\frac{2}{2n+1} - \frac{1}{n+1}\right)$$ Is there some general method for summing such series?
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1answer
51 views

Using Stolz Cesaro theorem to show $\lim(1^p+2^p+…+n^p)/(n^{p+1})=1/(p+1)$

I know Stolz-Cesaro theorem, and I'm supposed to use this to prove that $\lim(1^p+2^p+...+n^p)/(n^{p+1})=1/(p+1)$ So I made two sequences, $(x_n)$ which is: $\sum_{i=1}^n i^p$ And $(y_n)$ which ...
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1answer
74 views

Showing $\lim\frac{x_n}{y_n}=\lim\frac{x_{n+1}-x_n}{y_{n+1}-y_n}$

Let $(y_n)$ be an unbounded sequence of natural numbers with $y_{n+1}>y_n\forall n\in\mathbb{N}$. Let $(x_n)$ be another sequence, and suppose that $$\lim\frac{x_{n+1}-x_n}{y_{n+1}-y_n}$$ exists. ...
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1answer
108 views

The meaning of a definition involving multiple sums with Bernoulli numbers

Reading a paper regarding Bernoulli numbers, and I stumbled onto a definition. First let $$\frac{x}{e^x-1}=\sum_{k=0}^{\infty}B_k\frac{x^k}{k!}$$ The author then goes on to define new terms. Let ...
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2answers
67 views

Check the uniform convergence of $\sum\limits_{n = 1}^{\infty} \frac{n^2}{x} \exp(\frac{-n^2}{x})$

$$\sum\limits_{n = 1}^{\infty} \frac{n^2}{x} \exp{\frac{-n^2}{x}}$$ where $ 0 < x < \infty$ While $$\lim\limits_{n\rightarrow\infty} (a_n)^{1/n} = 0$$ The sum would converge. But how to check ...
4
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3answers
95 views

Show that $\lim_{n \to \infty} \frac{1}{n!} \frac{n^{n+1}}{e^n} = \infty$

Show that $$\lim_{n \to \infty} \frac{1}{n!} \frac{n^{n+1}}{e^n} = \infty$$ Wihout using stirlings aproximation to n! I've tried to compare this to a divergent sequence but didnt work. Also, I dont ...
8
votes
3answers
110 views

Any closed subset of $\mathbb C$ is the set of limit points of some sequence

Let $X$ be a closed subset of $\mathbb C$ with the usual norm. Prove that there exists some complex sequence $(u_n)$ such that $X$ is exactly the set of limit points of $(u_n)$. For the sake ...
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2answers
22 views

real analysis convergence of limit

Hi i have aquestion here which is whether $x_n = \frac{1}{n}$ for $n$ odd and $x_n = 1$ for $n$ even does the sequence converge? i am thinking of using the ratio test here. will it work ...
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0answers
34 views

Separable dual space implies existence of complete series

Let $\{x_n\}$ be a basis for a Banach space $X$ and let $\{f_n\}$ be the associated sequence of coefficient functionals. Prove or disprove: if $X^\ast$ is separable, then ${f_n}$ is complete ...
2
votes
1answer
31 views

Verification of identity $2\sum_{m,n\geq 1\,;\,m>n}\frac{1}{m^k n^k}=\left(\sum_{n\geq 1}\frac{1}{n^k}\right)^2 -\sum_{n\geq 1}\frac{1}{n^{2k}}$

Is this identity true? $$2\sum_{m,n\geq 1\,;\,m>n}\frac{1}{m^k n^k}=\left(\sum_{n\geq 1}\frac{1}{n^k}\right)^2 -\sum_{n\geq 1}\frac{1}{n^{2k}}$$ If so, how to prove it? Could you provide me a ...
8
votes
2answers
114 views

Prove that $\sum_{k=0}^\infty \frac{1}{16^k} \left(\frac{120k^2 + 151k + 47}{512k^4 + 1024k^3 + 712k^2 + 194k + 15}\right) = \pi$

How to prove the following identity $$\sum_{k=0}^\infty \frac{1}{16^k} \left(\frac{120k^2 + 151k + 47}{512k^4 + 1024k^3 + 712k^2 + 194k + 15}\right) = \pi$$ I am totally clueless in this one. Would ...
6
votes
2answers
35 views

Proving $\int_0^1 B_n(x) dx=0$ for Bernoulli polynomials

The Bernoulli polynomials $B_k(.)$ are given by $$ \frac{t\:e^{xt}}{e^t-1}=\sum_{k=0}^\infty B_n(x)\frac{t^n}{n!}, \quad |t|<2\pi. \tag{1} $$ I would like to prove that $$ \int_0^1 B_n(x) dx=0, ...
1
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1answer
16 views

Computation of conditionseries of real numbers

I am studying Riemann rearrangement theorem. and I stuck at the computation of this: $\sum\limits_{n=1}^\infty (\frac{1}{2n-1}-\frac{1}{4n-2}-\frac{1}{4n}) $ How to simplify to get the sum ? ...