0
votes
1answer
29 views

In the definition of sequences diverging to infinity, why must the constants be positive?

We are given the following definitions A sequence $(a_n)$ diverges to $\infty$ if for each $ M \in \mathbb{R}^+ \exists N_M \in \mathbb{N} \ \text{such that } \\ a_n > M \ \forall n \geq N_M$ ...
3
votes
7answers
115 views

Limit of sequences: $\lim \frac{(2n)!}{(n!)^2} $

Verify if the sequence $$\frac{(2n)!}{(n!)^2}$$ converges. My attempt: $$\frac{(2n)!}{(n!)^2} = \frac{(2n)(2n-1)...(n+1)}{n.(n-1)...1} \geq \frac{(n+1)^n}{n!} $$ Maybe it is easier to show that ...
2
votes
3answers
45 views

Proving that $\sum \frac{n^{n+1/n}}{(n+1/n)^n}$ diverges

Show that the series $$\sum \frac{n^{n+1/n}}{(n+1/n)^n}$$ diverges The ratio test is inconclusive and this limit is not easy to calculate. So I've tried the comparison test without success.
1
vote
1answer
21 views

$\sum r^n |\sin(nx)|$ convergence

Verify if the series $$\sum r^n |\sin(nx)|,\qquad r>0$$ Converges or diverges I've tried some comparisons with known series and the convergence tests, but didn't work. I think we ...
0
votes
0answers
36 views

convergence of a sequence of cauchy

I would like to ask something about the convergence of a Cauchy sequence in a space $X$ metric. There will be a metric space $X$ such that If $(x_n)$ cauchy sequence in $X$ then $(x_n)$ is not ...
0
votes
0answers
36 views

necessary condition on the sequence $\{a_n\}$ and $\{b_n\}$ such that $\sum a_n b_n$ converges [on hold]

Is there any necessary condition on the sequence $\{a_n\}$ and $\{b_n\}$ such that $\sum a_n b_n$ converges ?
1
vote
1answer
38 views

Viewing a sequence as a function on the space of positive integers

I see the following lines in a book : " Consider a bounded sequence of real or complex numbers $\{\eta_n\}$. Such a sequence $\{\eta_n\}$ defines a function $x(n) = \{\eta_n\}$ defined on the ...
1
vote
2answers
30 views

Question about point-wise convergent sequence of functions.

Let $$f_n(x)=n^2x(1-x^2)^n$$ be a sequence of functions on $[0,1]$. For $x=0$ and $x=1,$ clearly $f_n(x)=0$. Also for any $x_0$ in the open interval $(0,1)$, we have $0<1-x_0^2<1$. Therefore $$ ...
3
votes
4answers
238 views

Sequence proof (by induction, presumably) giving me trouble.

Let $a_1,...,a_n$ be a sequence of positive numbers. Show that $$(a_1+a_2+\cdots+a_n)\left(\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}\right)\geq n^2$$ Hint: Use the fact that for $x>0$ we ...
4
votes
2answers
119 views

Is there any closed form for the finite sum $1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+…+\dfrac{1}{n}?$ [duplicate]

I know that the infinite summation $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}+...$$ is divergent and also the sequence $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}-\ln ...
1
vote
2answers
102 views

Proof that $\dfrac{1}{e^x}=e^{-x}$ without converting it to $e^{x}e^{-x}=1$.

I want to show that $\dfrac{1}{e^x} = e^{-x}$ from the Taylor expansion of $e^x$. To express $\dfrac{1}{e^x}$ as a power series, I let: $$ \left(\dfrac{1}{0!}x^0 + \dfrac{1}{1!}x^1 + ...
0
votes
1answer
44 views

How to evaluate $\sum_{k=0}^{n} \alpha^k \binom{n}{k}$?

I am trying to show that the function that satisfies $f^\prime(x)=f(x)$ with $f(0)=1$ behaves in an exponential way (in other words, I want to justify writing it as $e^x$). I need to show that: $$ ...
2
votes
1answer
15 views

Limit of sum of series

Let $$\sum_{i=1}^\infty c_i^{(0)}$$ be a positive convergent series. Then the series: $$\sum_{i=1}^\infty c_i^{(1)}=\sum_{i=1}^\infty \ln(1+c_i^{(0)})$$ is also a positive convergent series. ...
-4
votes
1answer
34 views

Sequence converging to the supremum. [on hold]

let $T\subset \mathbb R$ let $a=\sup(T)$, and suppose that and $a<\infty$ and that $a \notin T$. Show that there exist a sequence $(a_n)\subset T$ for which $\lim_{n\to\infty}a_n=a$. If you are ...
0
votes
2answers
58 views

A identity relating a infinite series and a definite integral [duplicate]

Prove that, $$ \sum_{n=1}^{\infty} \frac{1}{n^n} = \int_{0}^{1} x^{-x}dx$$ I made no significant progress, I'm looking for hint/ideas to approach this problem. Thanks!
1
vote
2answers
33 views

sequence: use of stirling formula

I want to use the Sterling formula which says that: $lim_{n \rightarrow \infty}\dfrac{n!}{\sqrt{2*\pi}n^{n+1/2}*e^{-n}}=1$ I want to use it to show that $\lim_{n \rightarrow ...
4
votes
1answer
80 views

Sequence with equidistant terms

Consider the sequence $(u_{n})_{n \in \mathbb{N}}$ given by : $$ u_{0} \in \mathbb{Z} \quad \mathrm{and} \quad \forall n \in \mathbb{N}, \, \vert u_{n+1}-u_{n} \vert = 1.$$ Is the sequence ...
2
votes
0answers
31 views

rewriting the inverse image

If $\phi_k:\mathbb{R}^2\rightarrow \mathbb{R}$ are continuous functions, for all $k\geq0$ and $$\phi=\limsup_{n\rightarrow \infty }\phi_n$$ Let $A\subset \mathbb{R}$, is possible to write ...
1
vote
1answer
46 views

Is there a uniformly continuous function such that $a_{n+1} = f(a_n)$?

Let $a_{n+1} = a_n - a_n^2$ and $a_1 = \frac{2}{3}$. I already proved that $a_n \to 0$ Now I was asked, is there a uniformly continuous function such that $a_{n+1} = f(a_n)$? All I can think of is ...
1
vote
4answers
74 views

The series $\sum_n\Gamma (n-1/3)/(n-1)!$ diverges

I would like to prove that the series: $$\sum_{n=1}^{\infty}\frac{\Gamma (n-1/3)}{(n-1)!}$$ diverges. The problem is that I don't know how to begin. Intuitively I get the result, because observing ...
1
vote
0answers
23 views

A Question Regarding to a Series Comparison [closed]

If $0<a_i \leq 1 $ and $0<b_i \leq 1 $ for all $i\in \mathbb{N}$, and $$\sum_i a_i < \sum_i b_i $$ and $$\sum_i {a_i}^2 = \sum_i {b_i}^2 $$ Then does it follow that $$ \sum_i {a_i}^4 \leq ...
1
vote
1answer
36 views

Switching Limits and summation

I'm currently working on proving some theorems and there is one recurring problem that I somehow can't solve. $a_n$ is a real sequence in either $[0,1]$ or $\mathbb{R}$ that approaches $0$. ...
3
votes
1answer
67 views

$\lim (x_n-y_n)=0$ $\implies$ $\lim \Big(\dfrac {x_n}{y_n}\Big)=1$ ?

If $(x_n)$ , $(y_n)$ are sequences of non-zero real numbers such that $\lim (x_n-y_n)=0$ , then is it true that $\lim \Big(\dfrac {x_n}{y_n}\Big)=1$ ?
1
vote
2answers
58 views

Two questions about convergence in measure

I am currently studying for my analysis comprehensive exam and have a few questions about convergence in measure. First of all I know that for a sequence $\{f_n\} \in L^p(E)$, $1 \le p < \infty$, ...
1
vote
1answer
26 views

Convergence of Bounded Sequence Satisfying $2c_n \leq c_{n+1} + c_{n-1}$

Let $(c_n)$ be a bounded sequence satisfying $2c_n \leq c_{n+1} + c_{n-1}$. (a) Let $x_n = c_{n+1}-c_n$. Show that $(x_n)$ is increasing. (b) Show that $x_n \to 0$ as $n \to \infty$. ...
3
votes
1answer
21 views

Convergence of $\sum_{a\in \mathbb{N}^n} \left( \sum_{i=1}^n a_i^2\right)^{-\alpha}$

For which $\alpha$ (depending on $n$) does $$\sum_{a\in \mathbb{N}^n} \left( \sum_{i=1}^n a_i^2\right)^{-\alpha}$$ converge? Examples: For $n=1$ the series turns out to be $$\sum_{i=1}^\infty ...
3
votes
1answer
90 views

Calculate $\lim_{n\to\infty} \sum_{k=1}^{n} \Big(\frac{k}{n}\Big)^{\alpha k}$

Let $\alpha$ be a positive number. Calculate $$\lim_{n\to\infty} \sum_{k=1}^{n} \Big(\frac{k}{n}\Big)^{\alpha k}$$ Edit: I have deleted my attempt, it didn't seem to lead me anywhere and I discovered ...
3
votes
2answers
44 views

Differentiability for the uniform limit of a uniformly bounded sequence of functions

Let a sequence $\{f_n\}\subset C^1(\mathbb{R})$ and $f\in C(\mathbb R)$ such that $f_n \to f$ uniformly and $f_n, f'_n$ are uniformly bounded. Question : is $f \in C^1(\mathbb R)$ ?
1
vote
2answers
86 views

How to solve this and what is this number called? [duplicate]

What is the real number called to which the sequence $$\gamma_n =1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{n} - \log _e n$$ converges and what is the radius of convergence?
3
votes
1answer
41 views

If a series has the same sum under any rearrangement, then is it absolutely convergent?

Let $(V,\| \cdot \|)$ be a Banach space. Let $\{a_n\}$ be a sequence in $V$ such that $\sum a_n$ converges. Assume that for every bijection $f:\mathbb{N}\rightarrow \mathbb{N}, \sum a_n = \sum ...
0
votes
0answers
18 views

Identifying or bounding the zeros of the composition of two generating functions

Given two generating functions $$ G(a_n;x)=\sum_{n=0}^\infty a_nx^n \quad\text{ and }\quad H(b_n;x)=\sum_{n=0}^\infty b_nx^n, $$ what techniques are available for locating, or finding bounds on, the ...
1
vote
1answer
32 views

Question about prove of Absolute Convergence Test?

I'm self-studying from the book Understanding Analysis by Stephen Abbott and I have a question about the proof of the Absolute Convergence Test (theorem 2.7.6 on page 65). The author states that ...
0
votes
1answer
32 views

Relation between limit of functions and sequences

I need to prove that the sequence $$ \frac{\sqrt{n}}{\log n}$$ diverges. I know that $$\lim \frac{\sqrt{x}}{\log x} = \infty$$. Is there any theorem that relations the limit of the function with the ...
1
vote
1answer
47 views

Two sequences defined by recurrence relations satisfy $x_n/y_n<\sqrt{7}$ for all $n$

Let $(x_n)_{n\geq 1}$ and $(y_n)_{n\geq 1}$ be two sequences such that: $$x_{n+1}=x_n^2+1 \quad \text{ and } \quad y_{n+1}=x_n y_n$$ with $x_1=2$ and $y_1=1$ Prove that for all $n$ ...
-1
votes
0answers
25 views

Single formula sequence partitionining interval

So I have the real sequence for fixed $x\in \mathbb{R}$: $y_{j}(x)=\begin{cases}f_{j-1}(x) &\text{if } |f_{j-1}(x)|\leq |x_{j}|, \\ f_{j}(x) & \text{else}, \end{cases}$ where 1) ...
0
votes
1answer
32 views

A function relating $k$ and $j$, where $k=\max_{l\in \mathbb{N}}\sum_{i=0}^{l}2^{n-i}\leq j$ and $n= \lfloor \log_{2}j \rfloor$

Do you know any function that relates k and j, where $k=\max_{l\in \mathbb{N}}\sum_{i=0}^l 2^{n-i}\leq j$ and $n=\lfloor \log_2 j \rfloor$? So, say, for $j=3$: $n=1$ and $k=1$ because $3\geq ...
2
votes
4answers
82 views

For an irrational number $a$ the fractional part of $na$ for $n\in\mathbb N$ is dense in $[0,1]$

How to prove that the $\{$ fractional part of $n\alpha\mid n \in \mathbb{N}$ $\}$ is dense in $[0,1]$ for an irrational number $\alpha$. NOTICE that $n$ is in $\mathbb{N}$ Also notice that this is ...
1
vote
1answer
51 views

An arctan criterion for convergence?

Is the following inference correct, and if so, is it a mere curiosity? Let $\{a_k\}_{k \in \Bbb N}$ be a sequence of positive real numbers and set $$ P_k = \sum_{j=0}^{k-1} a_k \quad \mbox{and} ...
7
votes
1answer
117 views

Showing that $\sum_{i=1}^n \frac{1}{|x-p_i|} \leq 8n \left( 1 + \frac{1}{3} + \frac{1}{5} + \cdots + \frac{1}{2n-1} \right)$

I'm taking a summer analysis course and preparing for our final exam later this week. Our professor gave us the following problem on our mock exam, and I can't seem to get anywhere on it. Does anyone ...
1
vote
1answer
19 views

Making a sequence alternating

So I have the sequence $\{x_{j}=k+\frac{1+2(j-\sum_{i=0}^{k}2^{n-i})}{2^{n-k}}\}_{j\in \mathbb{N}}$, where $n= \lfloor log_{2}j \rfloor$ and k$=\{l\in \mathbb{N}:$maximum l s.t. ...
1
vote
0answers
32 views

A question on infinite series and boundedness of sequence

Let $(a_n)$ be a real sequence such that for every convergent real series $\sum x_n$ of positive terms , $\sum |a_n|x_n$ is also convergent , then is it true that $(a_n)$ is a bounded sequence ?
1
vote
0answers
83 views

A function equal to its Taylor series on an interval is also equal to its Taylor series on a subinterval with different center

Suppse the power series $ \sum_{n=0}^\infty a_n (x-a )^n$ has positive radius of convergence $R$ and thus defines a real analytic fuction $f$ on $(a-R,a+R).$If $x_0$ is a point with $|x_0-a|<R,$ ...
0
votes
3answers
43 views

If $a_n\ge nb_n$ and the sequence $(b_n)$ is unbounded, then the differences $a_{n+1}-a_n$ are also unbounded

Let $(a_n)_{n\geq 1}$ and $(b_n)_{n\geq1}$ be sequences of positive numbers such that $a_n\geq n b_n$ for all $n >1$. Prove that if $(a_n)_{n\geq 1}$ is increasing and $(b_n)_{n\geq 1}$ is ...
0
votes
2answers
42 views

Convergence of general Taylor series

For any $a,h \in \mathbb{R}$, how can we see the series, $\sum_{k=0}^{\infty} \dfrac{f^{(k)}(a)}{k!} h^k$ converges to $f(a+h)$?
4
votes
1answer
25 views

Uniform convergence of a sequence of polynomial logarithm

Let $P\in \Bbb{C}[X]$ of degree $d\ge 2$. For $n\in \Bbb{N}$ (include $O$). Denote by $P^n$ the $n$-th composition and $g_n: z\mapsto \frac{1}{d^n}\log(\max \{1,\vert P^n\vert\})$. Show that ...
3
votes
2answers
70 views

Prove that if an infinite series converges, then the associative property holds

I'm self-studying from the book Understanding Analysis by Stephen Abbott and have no idea how to do exercise 2.5.2 on page 57. The exercise is as follows: Prove that if an infinite series converges, ...
0
votes
1answer
38 views

Does every convergent sequence have a sub-sequence whose terms comes closer than any positive sequence?

Let $(x_n)$ be convergent sequence of real numbers and $(y_n)$ be any sequence of positive real numbers , then is it true that there is a sub-sequence $(x_{r_n})$ such that ...
4
votes
1answer
36 views

Is Banach space a correct context to study sequences and series?

Recently, I have reviewed elementary analysis and I realized that every theorem in the text(Rudin-PMA) about series can be generalized to Banach space. Here is an example. Below is the theorem ...
5
votes
3answers
356 views

Determining if a recursively defined sequence converges and finding its limit

Define $\lbrace x_n \rbrace$ by $$x_1=1, x_2=3; x_{n+2}=\frac{x_{n+1}+2x_{n}}{3} \text{if} \, n\ge 1$$ The instructions are to determine if this sequence exists and to find the limit if it exists. I ...
2
votes
0answers
49 views

$f\in C^\omega ((a-R,a+R),\mathbb{R})$ [closed]

We discuss the following question in the field of real numbers . A a power series $f(x)= \sum_{n=0}^\infty a_n (x-a )^n$ converges in $(a-R,a+R).$ Prove: $$\forall x_0\in\left(a-R,a+R \right), ...