1
vote
1answer
15 views

Convergent Sequence of Analytic Functions, Do Their Derivatives Converge?

If you have a sequence of real analytic functions that converge on every compact subset in your domain, do their derivatives necessarily converge to the derivatives of the function that they converge ...
3
votes
0answers
47 views

If $f_n(t):=f(t^n)$ converges uniformly to continuous function then $f$ constant

Let $f \colon [0,1] \to \mathbb R$ be continuous and let $f_n$ be defined by $$ f_n(x):=f(x^n), \qquad x \in [0,1], \,\, n \in \mathbb N. $$ Suppose there exists a continuous function $g$ on ...
1
vote
2answers
70 views

$\sum a_n$ converges, $a_n \in \mathbb{R}$, then there exists real sequence $b_n$ such that $b_n\rightarrow +\infty$ and $\sum a_n b_n$ converges.

Assume that $\sum a_n$ converges, $a_n \in \mathbb{R}$, then there exists real sequence $b_n$ such that $b_n\rightarrow +\infty$ and $\sum a_n b_n$ converges. Same to be easy at first thought, can ...
1
vote
1answer
27 views

find a $B_{n,j}$ such that $|A_{n,j}-L_j| \leq B_{n,j}$ $\forall n,j$ and $\sum_{j=0}^{\infty}B_{n,j}$ converges

We have $A_{n,j}= 3(-1)^j2^{n-j+1}\frac{(2(n-j)-4)!}{(n-j)!(n-j-2)!}\binom{j+2}{2}\frac{n^\frac{5}{2}}{8^n}$ and $L_j=(-\frac{1}{8})^j\binom{j+2}{2}\frac{3}{8\sqrt{\pi}}$ So I know $\lim_{n \to ...
0
votes
1answer
23 views

Is my proof about $\liminf$ of bounded real sequences correct?

Yesterday I asked in the question at A proof about the limit infimum of a bounded sequence about the proof of the following statement: Let $x_n$ be a bounded sequence of real numbers. Then the ...
5
votes
2answers
94 views

A series $\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}\frac{(i-1)! (j-1)!}{(i+j)!}H_{i+j}$ and $\zeta(3)$

We have $$ \sum_{j=1}^{\infty}\sum_{i=1}^{\infty} \displaystyle \frac{(i-1)! (j-1)!}{(i+j)!} H_{i+j} =\displaystyle 3 \: \zeta(3) $$ where $\displaystyle H_{n}:=\sum_{1}^{n} \frac{1}{k}$ are ...
3
votes
3answers
89 views

To prove the sum is convergent [duplicate]

Let$$a_n \ge 0$$ for all $n \in\Bbb N$. Show that if $$\sum_{n=1}^\infty a_n$$ converges, then $$\sum_{n=1}^\infty {\sqrt a_n\over n}$$ converges, too. The hint is to expand $$\left(\sqrt a_n-{1\over ...
4
votes
2answers
89 views
1
vote
1answer
37 views

A proof about the limit infimum of a bounded sequence

I tried to find a proof for this statement: If we have a bounded sequence $x_n$, then the limit infimum is defined as $a=\liminf_n x_n$ such that $a$ is the largest of real numbers which have the ...
0
votes
1answer
37 views

Do all series have a closed form representation of their partial sum? If not, can we feasibly prove that this is not the case?

The question was motivated by the way in which we approach the convergence and divergence of some series. During my undergraduate analysis course one of the only times in which the partial sum was ...
0
votes
1answer
50 views

The definitions of limit infimum and limit supremum

I have begun reading Rosenthal's "A First Look to Rigorous Probability Theory" and in order to reinforce my calculus background I am studying through the appendix section. Here he defined the limit ...
-1
votes
2answers
66 views

a question about sequence and series. prove $ \lim_{n \to \infty}( n\ln n)a_{n}=0$? [on hold]

Suppose $a_{n}>0$. $na_{n}$ is monotonic, and it approaches 0 as n approaches infinity. $\sum_{n=1}^{\infty} a_{n}$ is convergent. please prove $$ \lim_{n \to \infty} n\ln(n)\,a_{n}=0$$ I totally ...
0
votes
2answers
51 views

Supremum of $f_1(x)=1$ and $f_2(x)=x$

I'm trying to understand the supremem of a sequence of functions so I came up with a trivial case as follows - Let $(f_n(x))$ be a sequence of functions with $n$ having a value of either $1$ or $2$. ...
2
votes
2answers
79 views

Convergence of the series $\sum\limits_{n\geqslant1}(2-x)(2-x^{1/2})(2-x^{1/3})\cdots(2-x^{1/n})$

If $x >0$, consider the series $\sum\limits_{n=1}^\infty a_n$, where $$a_{n}=(2-x)(2-x^{1/2})(2-x^{1/3})\cdots(2-x^{1/n}).$$ Is it convergent? This question is in my textbook. I don't know how to ...
1
vote
2answers
40 views

Intuition for sequences of functions?

A sequence $(a_n)$ of real numbers can be thought of as a function that maps $\mathbb{N}$ to $\mathbb{R}$. The supremum of this sequence, if it exists, will be some $k \in \mathbb{R}$. A regular ...
0
votes
0answers
42 views

Proving pointwise convergence of Fourier series [on hold]

I'm trying to prove the pointwise convergence of a $2\pi$-periodic function $f$ to its Fourier expansion. The proof on my lecture notes stops at this formula: $$f(t)-P_{N,f}=\frac 1 {2 \pi} ...
2
votes
4answers
51 views

Prove a sequence does not tend to zero

Prove that the sequence $\dfrac{a^n}{2^nn^2}$ where $a>2$ does not tend to zero. I thought about writing $a=2+{\epsilon}$ then using binomial expansion which is valid for ${\epsilon}<1$ but ...
4
votes
3answers
109 views

Prove $ne^{-n}$ converges to zero

How would I prove that $ne^{-n}$ converges to zero? I've tried $ne^{-n}<{\epsilon}$ and then logging both sides but no further progress could be made. Thanks
3
votes
2answers
26 views

radius of convergence when root test fails

I'm stuck on this problem: Find the radius of convergence of $\sum \limits_{n=1}^\infty \frac{x^n}{((2+(-1)^n)^n} $ An attempt: From the root test, it seems $L$ does not exist: $$L= \lim_{n ...
4
votes
2answers
64 views

An asymptotic term for a finite sum involving Stirling numbers

The question is a by-product at the end of this post. The following asymptotic term will ensure the convergence of some series. $$ \frac{1}{n!} \sum_{k = 1 }^{n } \frac{{n \brack k}}{k+1} = ...
1
vote
1answer
28 views

Property of Conditionally Convergent series

If $ \sum a_n$ be an conditionally convergent series.For any real number R, is it true that there exists a sequence$\{b_n\}$ where each $b_i=1 $ or $-1$ such that $\sum a_nb_n$ converges to R?
4
votes
1answer
33 views

A question about the convergence of series.

If$\hspace{0.3cm}$ $\sum a_n$ and $\hspace{0.3cm}$$\sum b_n$$\hspace{0.3cm}$ are convergent then which of the following is true? $1.$$\hspace{0.3cm}$$a_{n+1}<a_n$$\hspace{0.3cm}$ $\forall n$ ...
2
votes
0answers
100 views

Find the function whose Taylor series is $\log(x)+\log(x+1)+\log(x+2)+\ldots$

How do I find a function $f$ whose Taylor series is $$\log(x)+\log(x+1)+\log(x+2)+\ldots$$ for some point $x=a$? It would seem that $$\left.\frac{\partial^n}{\partial x^n}f(x) \right|_{x=a} = ...
1
vote
0answers
23 views

Find limit property

Let $f$ be a function on $\mathbb{R}$ satisfy: $|f(x)-f(y)|\leq|x-y|$ $\forall x,y\in\mathbb{R}$. Consider the sequence: $$u_{n+1}=\frac{u_n+f(u_n)}{2},u_0=a$$ Research the limit property of this ...
0
votes
2answers
20 views

Prove for a monotone function, if $x_0$ is interior on interval $I$ then the one-sided limits exists

The proof goes like this: Lets suppose $f$ is nondecreasing (for nonincreasing we'll observe the function $-f(x)$). Let $x_0$ an interior point in the interval $I$, and $\left\{x_k\right\}$ an ...
1
vote
1answer
47 views

sum of the series $\sum_{k=0}^{\infty}(k+1)(1-|x_k|)|x_n|^k$ [closed]

Let $|x_n|<1$. Is the given series convergent or which conditions do we impose on the sequence $(x_n)$ to given series be convergent? $\sum_{k=0}^{\infty}(k+1)(1-|x_k|)|x_n|^k$
1
vote
1answer
19 views

Is $(\frac{1}{1-|x_n|})$ bounded or only bounded above if $|x_n|<1$? [closed]

Let $(x_n)$ be a sequence such that $|x_n|<1$. Is $(\frac{1}{1-|x_n|})$ bounded or only bounded above if $|x_n|<1$?
1
vote
1answer
48 views

Is $(|x_n|^{n+1})$ convergent or bounded if $|x_n|<1$?

Let $(x_n)$ be a sequence such that $|x_n|<1$. Is $(|x_n|^{n+1})$ convergent or bounded?
6
votes
2answers
268 views

A series with only rational terms for $\ln \ln 2$

We all know that $$ \ln 2 = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}. $$ Do you know a series with only rational terms for $$\ln \ln 2 = ?$$ Let's exclude base expansions with non ...
0
votes
0answers
52 views

sum of the series $\sum_{k=0}^{\infty}(k+1)(x_n)^k.$

Let $x_n$ be a sequence of real numbers such that $x_n\in(0,1).$ Find the sum of the series $\sum_{k=0}^{\infty}(k+1)(x_n)^k.$ My answer is $\frac{(x_n)'}{(1-x_n)^2}.$ But the term $(x_n)'$ make me ...
1
vote
0answers
26 views

Upper hemicontinuity and closed graphs

I have a problem with the definitions of upper hemicontinuity. In particular I found a picture that makes me wonder, not only about my understanding of this concept, but of the concept and application ...
2
votes
2answers
36 views

convergence of a sequence

I'm reading a paper, for proving a claim it defines $$ f_n(x) = \dfrac{(rx-x^2)^n}{n!} $$ when $ r = \frac{a}{b} $ is a rational, and $ I_n = \int^r_0 f_n(x) \cdot \sin x \cdot dx $ , and then it says ...
3
votes
2answers
33 views

What is the range of convergence of $\sum_{k=0}^\infty (k \cdot x \exp(-x))^k\cdot {1 \over k!} \cdot {1\over k+1}$?

I have the following series as an expression which occurs as a limit of a quotient of polynomials in $e$ and $x$ which I've expanded by polynomial long division into a series: $$f(x) = ...
0
votes
2answers
36 views

solving $|(x-3)(x-1)| $$\le$ $|\frac{1-x}{x-3}|$ graphicly [closed]

how to solve $|(x-3)(x-1)| $$\le$ $|\frac{1-x}{x-3}| $ in the graphic method?
1
vote
1answer
21 views

Sequence problem dealing with continuity and convergence.

I need help in this question. I figured out a way to solve the question but not sure the proof is valid. This is the question, Given $a \in\mathbb{R}$, and a function ...
1
vote
1answer
39 views

Interval of convergence

Find the interval of convergence of $\sum_{n=1}^{\infty }\frac {n^{2n}}{(2n)!}x^{n}$ I use ratio test and i found $\lim_{n\to\infty}|\frac{a_{n+1}}{a_{n}}|<1$ iff ...
1
vote
3answers
42 views

Series Convergence/Divergence $\frac{n^n}{(n+1)^{n+1}}$

Trying to establish whether $\sum x_n$ for $x_n := \frac{n^n}{(n+1)^{n+1}}$ converges or diverges. Here's what I've done so far: 1) n-th term: $x_n < \frac{n^n}{n^{n+1}} = \frac{1}{n}$, so ...
9
votes
8answers
255 views

Proof that if $a_1=1$ and $a_{n+1}=1+\frac{1}{1+a_n}$

Question: Proof that if $$a_n=\left\{ \begin{array}{ll} a_1=1\\ a_{n+1}=1+\frac{1}{1+a_n} \end{array} \right.$$ then $a_n$ converge, and then find $\lim_{n \to \infty}a_n$. I found ...
1
vote
2answers
85 views

Test for convergence $\sum_{n=1}^\infty \frac{1}{n}(\sqrt{n+1}-\sqrt{n-1})$

Test for convergence $$\sum_{n=1}^\infty \frac{1}{n}(\sqrt{n+1}-\sqrt{n-1})$$ So far I attempted to use the ratio test, but I'm stuck on what to do after. ...
1
vote
0answers
54 views

Backward from infinity?

The following question has been raised and answered lately: Problem 6 - IMO 1985 Please take a look at the Reverse method part of the answer given by this author. What's happening there is that we ...
0
votes
1answer
16 views

Multiplication of non-summable sequences convergent to $0$

Let $(\lambda_n)_n$ be a sequence of real numbers which converges to $0$ (i.e., is in $c_0$), but is not in $\ell^p$ for any $1\leq p<\infty$, e.g., $\lambda_n=\frac{1}{\log(n+2)}$ for ...
0
votes
0answers
17 views

Limit of a sum over an increasing finite set: dominated convergence / riemann integral

Let $(S_k)_{k\in\mathbb{N}}$ be a sequence of finite sets where $S_k \subset [0,1]$ for all $k$. It is assumed that $S_k\subset S_{k+1}$ and that $$\lim_{k\to\infty }\max_{s\in S_k} (s-s')=0$$ where, ...
9
votes
2answers
108 views

Divergent of a vector field on a sequence of spheres

I'm studying for my exams and I found this problem in the book "Advanced Calculus", written by Friedman: "Consider a sequence of spheres $S_n$ in $\mathbb{R}^3$ with center $P_n$ and radius $r_n$, ...
3
votes
1answer
62 views

Limit of $x_n = 0.5(x_{n-1} + x_{n-2})$ - help finishing proof…

EDIT: Fixed geometric proof off-by-one error. I am looking for the limit of $x_n = 0.5(x_{n-1} + x_{n-2})$ with $x_2 > x_1$ arbitrary. I can show $\forall n: |x_{n}-x_{n+1}| = \frac{c}{2^{n-1}}$ ...
1
vote
2answers
76 views

Taylor series of the function $f(x) = (1+x) ^{\frac{1}{x}}$

Good night!! I got this problem and I'd like to find all the mistakes in my statement. This is my prblem. Find the binomial coefficients of $f(x) = (1+x)^{\alpha}$, with $\alpha \in \mathbb{R}$, and ...
3
votes
6answers
81 views

Prove that $\sum\limits_{n=0}^{\infty}{(e^{b_n}-1)}$ converges, given that $\sum\limits_{n=0}^{\infty}{b_n}$ converges absolutely.

It's a question from a test that I had, and I don't know how to prove this, so I am forwarding this to you. $\sum \limits_{n=0}^{\infty }\:b_n$ is absolutely convergent series . How to prove that ...
4
votes
4answers
151 views

Convergence of $\left(1+\frac{1}{n}\right)^n$, given that $\left(1+\frac{1}{2^n}\right)^{2^n}$ converges

I know how to prove that the sequence $$\left(1+\frac{1}{2^n}\right)^{2^n}$$ has a limit. Can I use this knowledge to quickly get the fact that $$\left(1+\frac{1}{n}\right)^n$$ also has a limit?
6
votes
1answer
104 views

The sequences $x_n$ and $y_n$ converges

Let $\quad2{x}_{n+1}=1+{y}^{2}_{n},\quad 2{y}_{n+1}=2{x}_{n}-{x}^2_{n},\quad n\in\mathbb{N};\quad 0\leq {y}_{0}\leq \frac{1}{2}\leq {x}_{0}\leq 2.$ Prove that the sequences $ \begin{Bmatrix} ...
1
vote
0answers
27 views

Exchanging Limits in Series inversion

I have the Lagrange Bürmann formula as follows: $$\sum^{m-1}_{n=1}\frac{1}{n!}(w-b)^n\lim_{z \rightarrow a} \frac{d^{n-1}}{dz^{n-1}} \left(f^{'}(z)\frac{z-a}{g(z)-g(a)}\right)^n$$ Where g(z) is the ...
0
votes
0answers
50 views

Show $\sum_{k=1}^\infty a_k <\infty$ implies $\frac{1}{n} \sum_{k=1}^n k a_k \rightarrow 0$ [duplicate]

Suppose $$\sum_{k=1}^\infty a_k <\infty$$ Prove that $$\frac{1}{n} \sum_{k=1}^n k a_k \rightarrow 0$$ as $n \rightarrow \infty$. I tried Stolz-Cesaro, and also breaking up the sum into two parts, ...