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

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1
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0answers
68 views

Show that for almost all $x$ in $[-1,1]$, the series $ \sum\limits_{n=1}^\infty \frac{1}{2^n \sqrt{|x-r_n|}} $ converges

Let $\{r_n\}$ be a sequence of real numbers in $[-1, 1]$, then show that for almost all $x$ in $[-1,1]$, the series $$ \sum_{n=1}^\infty \frac{1}{2^n \sqrt{|x-r_n|}} $$ converges. I am struggling on ...
0
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1answer
18 views

Statistics for Sports League Qualification

How can one quantify and predict the needed points for qualify in a league given an up-to-date results registry? For instance, regarding Basketball Euroleague, there's 8 teams in a league with direct ...
1
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2answers
35 views

Confusion about convergence tests

There are two problems in particular I'm having trouble with. 1.) $$\sum_{n=1}^\infty(-1)^n\frac{\ln(n)}{\sqrt{n}}$$ I'm having difficulty demonstrating that $\frac{\ln(n+1)}{\sqrt{n+1}} \le \frac{\...
4
votes
1answer
630 views

A necessary condition for series convergence with positive monotonically decreasing terms

Suppose that the series $\sum a_n$ is convergent and the terms are positive and decreasing. Is it necessary that $\lim n \log n \, a_n$ exists and $$\lim_{n \to \infty} \,\,n \log {n} \,a_n = 0.$$ ...
0
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0answers
27 views

A little question about the fact that $(\frac{1}{a_n}) \to +\infty$ given a certain condition

$(a_n)$ is a null sequence and $a_n > 0$ for all $n \implies (\frac{1}{a_n}) \to +\infty$. Proof: For given $C$ there's some $N$ such that $n > N \implies |a_n| < \frac1C \implies 0 < ...
0
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1answer
60 views

if $f(n) \to g(n)$ then $\frac{\sum f(n)}{\sum g(n)} \to 1$?

Say we have some function $f(n)$ that behaves like $g(n)$ for large $n$. It is easier to analyse in general the $g(n)$. Then it seems intuitive to say $$\frac{\sum f(n)}{s_n}\to 1$$ where $s_n= \sum g(...
0
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1answer
68 views

Find the formula for the..

For the following, find the formula for the $n^{th}$ term of the sequence $$1,3,6,10,.........................$$. My Attempt: $$t_1=1=(1-1)+1$$ $$t_2=3=(2-1)+2$$ $$t_3=6=(3-1)+3$$. The answer in ...
4
votes
0answers
58 views

Intuition about the Bernstein polynomials proof of the Weierstrass approximation theorem

The Weierstrass approximation theorem can be stated as follows: Let $f\in C([a,b])$, then there exists a sequence $(p_n)_{n\in \mathbb{N}}$ of polynomials in $[a,b]$ such that $(p_n)$ converges ...
1
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0answers
38 views

Is this a usual approach to uniform convergence?

Consider $E\subset \mathbb{R}$ and a sequence of functions $(f_n)_{n\in \mathbb{N}}$ with $f_n : E\to \mathbb{R}$. The easiest form of convergence we can define is the pointwise convergence - we say ...
1
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4answers
34 views

The right way to cancel out the terms in the following telescoping series

So how do I cancel and simplify the terms in the following telescopic series. Been at it for hours, cant seem to figure it out. $\sum\limits_{k = 1}^n \frac{1}{2(k+1)} -\frac{1}{k+2}+\frac{1}{2(k+3)} ...
1
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0answers
65 views

Closed form representation of an exponential series

Let $a\in\mathbb{R}$, $t\in\mathbb{R}\ge 0$ and consider the following series $$ f(t)=\sum_{n=1}^{\infty}\frac{\exp({-a^2 t})-\exp({-n^2 t})}{n^2-a^2} . $$ The series is well defined in the poles at ...
0
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0answers
19 views

Asymptotic Expansion for Function with an Embedded Integral [duplicate]

So I'm trying to find the asymptotic expansion as $x \to \infty$ of: $$f(x)=\frac{1}{\bigg[A-\int_{x_0}^x\frac{\lambda^y}{\Gamma(y+1)}dy\bigg]^{\frac{1}{\alpha}}}$$ where $x_0>0$ and $\alpha>0$ ...
1
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1answer
59 views

Convergence of $ \sum_{n=1}^\infty\frac{e^nn!}{n^n} $ [closed]

Is the following series convergent? $$ \sum_{n=1}^\infty\frac{e^nn!}{n^n} $$
3
votes
1answer
50 views

Prove that if $A$ is nonsingular, then the sequence $X_{k+1}=X_k+X_k(I-AX_k)$ converges to $A^{-1}$ if and only if $ρ(I-X_0A)<1$. [closed]

Prove that if $A$ is nonsingular, then the sequence $X_{k+1}=X_k+X_k(I-AX_k)$ where $A$ and $X_k$ are $n\times n$ matrices with $k=0,1,2,...$ converges to $A^{-1}$ if and only if $ρ(I-X_0A)<1$. I'...
2
votes
2answers
85 views

limit of $|n^t\sin n|$

It is known that $\{\sin n : n\in\mathbb{N}\}$ is dense in $[-1,1]$, hence $\lim_{n\to\infty}\sin n$ doesn't exist and also $\lim_{n\to\infty} n^t\sin n$ doesn't exist for all $t>0$ (the reason is ...
-1
votes
3answers
51 views

Partial fraction decomposition: $f(x) = \frac{x}{1-x-x^2}$

Consider the function $$f(x) = \frac{x}{1-x-x^2}$$ (a) Determine a recursive formula for the coefficients $c_n$ of the Maclaurin series of $f$. (b)Using the partial fractions decomposition of $ f(x)$...
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4answers
486 views

Doubt on limit of sum of sequences. Two procedures leads to different answers.

I have the following problem. Determine the convergence or divergence of the sequence $(x_n)$ where $$x_n=\frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n}.$$ My first approach was: Well, ...
0
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1answer
32 views

Confused about the choice of $\epsilon$ for proving sequences are not null

Definition of null sequence I am using: $(a_n) \to 0$ as $n \to \infty \iff$ given $\epsilon > 0$, there's $N$ such that $n > N \implies |a_n| < \epsilon.$ The sequences below are not null ...
0
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4answers
78 views

Proving convergence of $\sum_{n=1}^\infty \frac{1}{n} \log(1 + \frac{1}{n})$

I wonder if this is a valid proof that $$\sum_{n=1}^\infty \frac{1}{n} \log\left(1 + \frac{1}{n}\right)$$ converges: $$\frac{1}{n} \log\left(1 + \frac{1}{n}\right) < \frac{1}{n} \log(1) = \frac{1}{...
3
votes
3answers
119 views

What is a mathematical expression for the sequence $\{1,1,-1,-1,1,1,-1,-1,\dots\}$?

What is a mathematical expression for the sequence $\{1,1,-1,-1,1,1,-1,-1,\dots\}$, that is $1$ and $-1$, two at a time alternating?
1
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2answers
55 views

How can I prove that $\sum\limits_{i=1}^\infty \frac{1}{(2i-1)^{2}} = \frac{π^2}{8}$?

I have the series $$\sum\limits_{i=1}^\infty \frac{1}{i^{2}} = \frac{π^{2}}{6}$$ How can I prove that: $$\sum\limits_{i=1}^\infty \frac{1}{(2i-1)^{2}} = \frac{π^2}{8}$$ I have been looking for it ...
1
vote
1answer
27 views

Evaluate if series with exponential diverges or converges

The task is to evaluate for what values of $a \in \Bbb R_+$ does the series $$\sum_{n=1}^\infty \frac{a^n \times n!}{n^n}$$ converge. I've already checked with the ratio test that it converges for $ a ...
4
votes
4answers
46 views

Convergence of $r^n/n$ when $|r| > 1$.

Consider $r\in \mathbb{R}$ and the real number sequence $(a_n)_{n\in \mathbb{N}}$ where $$a_n = \dfrac{r^n}{n}.$$ If $|r|<1$, we know that $r_n\to 0$ when $n\to \infty$. Since $1/n \to 0$ when $n\...
2
votes
1answer
33 views

Integral inequality for a Cauchy exponential series product

My goal is to get an inequality $\forall t>0$ for the following integral $$ \int_0^t \left(\sum_{n=1}^\infty \exp(-n^2 t_0)\right)^2\,\mathrm{d}t_0 \le f(t). $$ The goal is to at least lose the ...
1
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1answer
30 views

Prove the sequence $2, 1, 0, -1, -2, -3, -4, \ldots$ doesn't tend to $0$

Here's the definition of null sequence $(a_n) \to 0 \iff \forall \epsilon > 0, \exists N $ such that $n > N \implies |a_n| <\epsilon.$ reworded as $ \forall \epsilon > 0, \exists N \...
1
vote
2answers
76 views

Find all the points which makes the series normal converges and uniformly converges

I'm learning "Complex Analysis", section Series and Convergence, and I got stuck on this problem (actually, just a small part of this problem): Find all the values of $z$ which makes this series ...
3
votes
1answer
83 views

Sequence of continuous function converging pointwise to Thomae's function

Recall that Thomae's function (also called Popcorn function) $f\colon\mathbb{R}\to\mathbb{R}$ is defined as $$ f(x) = \begin{cases} \frac{1}{q} & \text{ if } x=\frac{p}{q} \neq 0 \text{ is ...
0
votes
2answers
39 views

How to show that if $f_{k}(x)=kxe^{-kx}$, {$f_k$} converges to the zero function?

$f_{k}(x)=kxe^{-kx}$ for all $x\ge0$ and $k\ge1$, how to show {$f_k$} converges to the zero function on $[0,\infty)$ pointwise? By L'hospital's Rule we can show that: $\lim_{x\to\infty}xe^{-x}=0$. ...
2
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0answers
44 views

Fundamental theorem of arithmetic, prove that these matrices are the same apart from the main diagonal.

I know and I can prove that the fundamental theorem of arithmetic is encoded uniquely by this recurrence written in Mathematica 8: ...
17
votes
4answers
796 views

Prove that $n^{n+1} \leq (n+1)^{n} \sqrt[n]{n!}$

Let $n$ be a positive integer. I conjectured that the following inequality is true \begin{equation} n^{n+1} \leq (n+1)^{n} \sqrt[n]{n!} . \end{equation} Anyhow I could neither prove nor disprove it. I ...
2
votes
2answers
38 views

A sequence of function $\{f_n\}$ that converges to $f$ point-wise almost everywhere

I want to construct a sequence of function $\{f_n\}$ that does NOT converge point-wise in a domain $D$ but converges point-wise almost everywhere in $D$. Define: $$f_n:D=[0,1]\cup\{2,3\}\to \mathbb ...
1
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3answers
62 views

Prove that $\lim_{n\to\infty} \frac{\log n}{n} = 0$

I want to show the above problem I already write in the title by using the following equality: $\lim_{n\to\infty} \frac{a^n}{n} = \infty$ if $a>1$. I tried to solve this problem by seeing the ...
0
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1answer
38 views

Find the sum to the $n^{th}$ term of the series

Find the sum of $n^{th}$ term of the series: $$\frac{3}{1^2.2^2}+\frac{5}{2^2.3^2}+\frac{7}{3^2.4^2}+\cdots$$ I could not find the rule for the $n^{th}$ term.
1
vote
1answer
24 views

How to show $\sum\limits_{i=1}^{t}\frac{1}{i}2^{t-i}=2^t\ln 2 -\frac{1}{2}\sum\limits_{k=0}^\infty \frac{1}{2^k(k+t+1)}$

How to show the below equation ? $$\sum\limits_{i=1}^{t}\frac{1}{i}2^{t-i}=2^t\ln 2 -\frac{1}{2}\sum\limits_{k=0}^\infty \frac{1}{2^k(k+t+1)} ~~~~~(t\in \mathbb Z^+)$$
3
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2answers
82 views

How to show convergence of a particular series from first principles

I am refreshing my techniques for series and am currently stuck with the following exercise: Show that for every real number $x > 1$ the series $$ \sum_k \frac{2^k}{1 + x^{2^k}} $$ ...
3
votes
2answers
30 views

If $f(x)=\lim_{n\to\infty}[2x+4x^3+\cdots+2nx^{2n-1}]$, $0<x<1$, then find $\int f(x)\mathrm{d}x$

If $f(x)=\lim_{n\to\infty}[2x+4x^3+\cdots+2nx^{2n-1}]$, $0<x<1$, then find $\int f(x)\mathrm{d}x$ $$f(x)=\lim_{n\to\infty}2x[1+2x^2+\cdots+nx^{2n}]$$ $$S=\frac{f(x)}{2x}=1+2x^2+\cdots+nx^{2n}$$...
4
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2answers
51 views

Simplify the series given by the recurrence relation $na_n=2a_{n-2}$

If you are given a recurrence relation such that: $$na_n=2a_{n-2}\implies a_n= \begin{cases} 0 & \text{odd} \,n \\ \frac{2}{n}a_{n-2} & \text{even} \,n \end{cases}$$ My textbook suggests ...
0
votes
1answer
60 views

Suppose every subsequence of X converges to 0. Show that lim(X)=0

Question: Suppose that every subsequences of $X = (x_n)$ has a subsequence that converges to $0$. Show that $\lim(X) = 0$ My attempt: Suppose that $\lim(X) =L \neq 0$ Let $\epsilon > 0$, and ...
2
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0answers
23 views

Which error does one usually need to consider in numerical analysis?

When analyzing performance of a numerical method, I have considered and plotted $$\| x_n - x^* \|$$ where $x_n$ is the $n$-th iteration and $x^*$ is the true value (in our toy examples, it wasn't hard ...
0
votes
0answers
31 views

If $\forall \varepsilon>0\;\;\;\;a(n)\leq C(\varepsilon)\; n^{r+\varepsilon} $ then $\sum_{n=1}^{\infty}\frac{a(n)}{n^s}$ converge

Let $\{a(n)\}_{n\in\mathbb{N}}$ be a sequence of real numbers. I tried to prove that if $$\forall \varepsilon>0\;\;\;\;a(n)\leq C(\varepsilon)\; n^{r+\varepsilon} $$where $C(\varepsilon)$ is a ...
2
votes
1answer
30 views

Pointwise convergence of $(f_n)$ on $I$ to a function $f$

Suppose that a sequence of functions $(f_n)$ defined on an interval $I$ satisfies $|f_n(x) − f_n(y)| \leq |x − y|$ for any $x, y ∈ I$ and any $n \geq 1$ and $(f_n)$ converges pointwise on $I$ to ...
5
votes
1answer
76 views

Reference request for an identity involving binomial coefficients

The identity is $$\sum_{i\ge 0}(-1)^i \binom{\frac{s^k + s^{-k} - 10}{4}}{i}\binom{\frac{s^k + s^{-k} - 10}{4}+4}{\frac{gs^k + 2g^{-1}s^{-k} - 4}{8}-i}= 0$$ where $k\gt 1$ , $s = 3+2\sqrt{2}$ ...
0
votes
0answers
51 views

Sum of the reciprocal of tetration?

Let $$f(x)=\sum^\infty_{n=1}\frac{1}{{}^xn}$$ where ${}^xn$ is n tetrated to the xth. What are f(2) and f(3), and could you please also explain how you reached these answers?
2
votes
2answers
60 views

Using the Ratio Test on Trig Functions

I've got to use the Ratio Test to determine whether this series is convergent or divergent: $$\sum_{n=1}^\infty \frac{cos(n\pi/3)}{n!}$$ Taking the $\lim \limits_{n \to \infty}|\frac{a_n+1}{a_n}|$ ...
0
votes
1answer
28 views

How do we prove a sequence doesn't tend to $+\infty$?

Suppose we want to prove that the sequence $1, 1, 2, 1, 3, 1, 4, 1, 5, 1,\ldots$ doesn't tend to $+\infty.$ Here's what I think we might do: We negate the definition that says $(a_n) \to +\infty ...
0
votes
1answer
21 views

Cauchy Convergence criterion

Use the Cauchy convergence criterion to show that the following sequence convergences (without quoting results for the series representation of the transcendental number $ e $). $$ \sum_{k=1}^n \...
10
votes
1answer
155 views

A problem involving the product $\prod_{k=1}^{n} k^{\mu(k)}$, where $\mu$ denotes the Möbius function

Let $\mu$ denote the Möbius function whereby $$\mu(k) = \begin{cases} 0 & \text{if $k$ has one or more repeated prime factors} \\ 1 & \text{if $k=1$} \\ (-1)^j & \...
1
vote
3answers
30 views

Meaning of the numbers in a sequence definition

The sequence $(a_n)$ tends to $+ \infty \iff$ given any number $C$, there's a number $N$ such that $n > N \implies a_n \ge C.$ Given a certain $N$ it's not difficult to prove the implication, but ...
0
votes
1answer
86 views

limit of sum with binomial coefficients

I have the problem to compute next double sum \begin{equation} \sum_{n=2}^{\infty}\frac{(-1)^n}{n-1}\sum_{k=0}^{n}(3n-k)^j{n\choose k}A^{n-k}B^k\;, \end{equation} being $j\gg1$ an integer number and $...
3
votes
2answers
54 views

Which integer sequence starts with small elements, and stays there for a (really) long time, but eventually escapes the initial area?

Graphically, I am searching for something like this: The only additional requirement would be that the elements are defined by a closed formula or "simple" recursion, i.e. no definition by cases (...