1
vote
2answers
44 views

a question about limit, I am struggling with this!

Suppose that {$a_n$}is a sequence of positive numbers.For each n which is a natural number,let $b_n$=($a_1+a_2+......a_n$)/n,prove that $\sum b_n$ diverges to $+\infty$. This question is my homework, ...
0
votes
0answers
16 views

Absolute convergence does not implies convergence

Find a space and a series that converges absolutely but it does not converges. It is clear that the space can't complete or Banach.
1
vote
0answers
26 views

Prove that the limit of the function a sequence is the same as the function of the limit of the sequence [duplicate]

Assume that $\lim_{n\to \infty}a_n=a.$ Suppose that the function $f$ is continuous everywhere including at $a$. Form the sequence $(f(a_n))_{n=1}^{\infty}$. Prove that $\lim_{n\to ...
0
votes
0answers
16 views

ratio test for convergence of series, different versions

In lecture we had the ratio test: Let $(a_k)$ be a sequence in $\mathbb{K} \in\{ \mathbb{R}, \mathbb{C}\}, a_k\not= 0$ for all $k \ge k_0$, where $k_0\in \mathbb{N}$. (I) If there is a $q\in (0,1)$, ...
3
votes
1answer
108 views

How was this sequence discovered?

Let $N$ be a positive integer and consider the following rational sequence for $n \ge 0$: $$ a_{n+1} = \frac{N a_n + N}{a_n + N}, a_0 \in \Bbb{Q}. $$ If $-\sqrt{N} < a_0 < \sqrt{N}$, then ...
0
votes
1answer
27 views

I need to show that this sequence is increasing and I'm almost there but I need help on last step.

Let $(1+\frac{1}{n})^n$ be a sequence and $f(x)=(1+\frac{1}{x})^x $ on $[1,inf)$. I need to show that f is non-decreasing by showing that $f'(x)\ge0$. So far I have: Let $g(x)=ln(f(x))$, where $ln$ ...
1
vote
1answer
74 views

Prove that $r^n/n!$ converges where $n\ge r$ [on hold]

The answer is in the title of the question. I need to show it converges to 0 and $r>0$. I am sorry if this is a bad question, I'm having trouble explaining it. So essentially this Do the ...
4
votes
1answer
82 views
+50

Prerequisites for understanding G.H. Hardy's 'Divergent Series'

I picked up a copy of G.H. Hardy's 'Divergent Series' a few days ago. So far I love it, as I love the ideas associated with sequences and series, but I am finding it a bit difficult to understand. I ...
3
votes
4answers
115 views

Is $\sum_{n=1}^\infty {1\over 3^{\sqrt{n}}}$ convergent?

Is $\sum_{n=1}^\infty {1\over 3^{\sqrt{n}}}$ convergent ? I use it to compare with $1/n^2$, and then I used LHôpitals rule multiple times. Finally , I can solve it. However,I think we have other ...
1
vote
1answer
40 views

a question about convergence of sequecce!I have tried cauchy method, but it doesn't work

suppose $a_n>0$,and$\sum_{i=0}^\infty a_i$ is convergent,so we need to prove $\sum_{n=1}^\infty{ {1\over n}(a_n+a_{n+1}+\cdots+a_{2n})}$ is also convergent! I have tried cauchy method, but maybe ...
0
votes
1answer
24 views

Minimisation of Finite sum of a decreasing sequence

If $a_{1}<a_{2}<a_{3}<...<a_{n}$, find the minimum value of $$\sum_{i=1}^{n} (x-a_i)^{2}$$ Then find the value of $$f(x)=\sum_{i=1}^{n} |x-a_i|$$ Hi all, what would the best way be ...
0
votes
0answers
26 views

Find an analytic function [duplicate]

Is it possible to find an analytic expression for a smooth, continuous, single-variable function $y=f(x)$ such that: 1) $f(0) = 0$ 2) $f(x_1) = y_1$ 3) $f(x) > 0, \forall x>0$ 4) ...
0
votes
2answers
29 views

Prove that the convergence of the sequence (s3n) implies the convergence of (sn).

I write $s_n-s$, as $(s_n^3-s^3)/(s_n^2+s_n*s+s^2)$, true for all $n>N$. I'm trying to show that the denominator is convergent. But I don't know how to do this. Need help! Thanks. (Sorry about ...
0
votes
1answer
27 views

Prove that the convergence of the sequence (sn) implies the convergence of (s3n) [on hold]

Case 1: $s>0$. Assume $s>0$. Then there exists $N$ such that for all $n>N$ $s_n>0$. If $s_n^3-s^3$ converges to $0$, then write $s_n-s$, as was done in OH, as ...
7
votes
0answers
92 views

Problem 6 - IMO 1985

For every real number $x_1$ construct the sequence $x_1,x_2,x_3,\ldots$ by setting $x_{n+1}=x_n(x_n+\frac{1}{n})$ for each $n \ge 1$. Prove that there exists exactly one value of $x_1$ for which $0 ...
0
votes
0answers
20 views

Convex Feasibility problem on Hilbert space

Let $H$ be a Hilbert space with real inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$. Consider a closed convex set $X \subset H$ and let $P_X: H \rightarrow X$ be the ...
2
votes
2answers
77 views

Series $\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$

Does anyone know the sums of the following two series? $$\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$$ $$\sum_{n=1}^\infty (-1)^{n+1} \frac{x^{4n-1}}{4n-1}$$ I encounter such series in my work.
2
votes
2answers
75 views

What is the sufficient and necessary condition for changing the order of summation?

What is the necessary and sufficient condition for $\sum\limits_{i=0}^{\infty }{\sum\limits_{j=0}^{\infty }{{{a}_{ij}}}}=\sum\limits_{j=0}^{\infty }{\sum\limits_{i=0}^{\infty }{{{a}_{ij}}}}$? Suppose ...
0
votes
1answer
13 views

Uniform Convergence of Series Help

Suupose the sequence $(b_k) , k\geq 0$ satisties $\sum k|b_k| < \infty$, then show that $\sum_{k=0}^\infty b_kx^k$ converges uniformly to a function $g$ on $|x| \leq 1$ and that $g'(x) = ...
1
vote
2answers
66 views

infinitely descending natural numbers

Show that there is no infinitely descending sequence of natural numbers. I was thinking that there exists no infinite descending chain on the natural numbers, since every chain of natural numbers has ...
0
votes
1answer
24 views

$\sum_{n=1}^{\infty}|a_{n} x^{n}| < \infty \implies \sum_{n=1}^{\infty} |a_{n}| (|x|^{n-1}+ |x^{n-2}y|+…+|y^{n-1}|) <\infty $?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$. Now, we let, $x, y \in \mathbb R$ with $x\neq ...
1
vote
3answers
43 views

$\sum_{n=1}^{\infty} a_{n}= \sum_{n=1}^{\infty}b_{n}$ and $\sum_{n=1}^{\infty} |a_{n}| < \infty \implies \sum_{n=1}^{\infty} |b_{n}| < \infty$?

Suppose that $\{a_{n}\}_{n\in \mathbb N}, \{b_{n}\}_{n\in \mathbb N} \subset \mathbb C$ such that both the series, $\sum_{n=1}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} b_{n}$ converges, and its sum ...
1
vote
1answer
44 views

How to justify, $\sum_{n=1}^{\infty} a_{n} x^{n} - \sum_{n=1}^{\infty}a_{n}y^{n}=\sum_{n=1}^{\infty} a_{n} (x^{n}-y^{n})$?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$ and we let $K_{1}$ be a compact subset of ...
1
vote
1answer
30 views

Limit of sequence that is root of sums of powers

Suppose $z_1,\ldots,z_k$ are complex numbers with $|z_1|>\cdots>|z_k|$, and let $c_1,\ldots,c_k$ be non-zero complex numbers. Let $a_0,a_1,\ldots$ be the sequence defined by $$a_n=\sum_{i=1}^k ...
1
vote
3answers
69 views

Question on Uniform Conergence

I need to show that $\sum_{k=1}^\infty$$(\frac {x}{2})^k$ does not converge uniformly on (-2, 2) I know I have to show that $\sup_{x\in X}\lvert f_n(x)-f(x)\rvert\nrightarrow0 $ as ...
0
votes
0answers
25 views

Construct a Converging Series from the Following

This is more of a request for advice than a request for solution. Last night we were given the following and nobody figured it out in the time given (about 5 minutes). I think this is a problem many ...
2
votes
2answers
45 views

Is it true that, $|e^{x}-e^{y}|\leq C \cdot |x-y|$?

Define $f:\mathbb R \to \mathbb R$ such that $f(x)= e^{x}-1:= \sum_{n=1}^{\infty} \frac{x^{n}}{n!};$ for $x\in \mathbb R.$ My Question: Can we expect $|f(x)-f(y)|\leq |x-y| \cdot C;$ where $C$ is ...
0
votes
4answers
43 views

Determine if the given sequence converges or diverges

Let $(x_n)$ be a sequence defined as $x_n = \frac{1}{n} \sum_{j=1}^{n} \frac{j+1}{j^2}$ . We want to know if $(x_n)$ converges. The trouble I am having here is that the sum depends on $n$. We know the ...
0
votes
1answer
60 views

Is this construction already a contradiction

Let's say we have $\ell^p$ for $p>2$ and $\ell^2 \subsetneq \ell^p$. Let $U$ be a closed subspace of $\ell^2$. Is it possible that $U$ is isomorphic to $\ell^p$ as a Banach space?- I hardly think ...
1
vote
1answer
45 views

$\lim\limits_{n \to \infty} \sup_{k \geq n} (\frac{1+a_{k+1}}{a_k})^k \ge e$ for any sequence $\{a_n\}$ with positive terms

Show that $$\lim_{n \to \infty} \sup_{k \ge n} \left(\frac{1+a_{k+1}}{a_k}\right)^k \geq e$$ for any sequence $\{a_k\}$ with positive terms, and that this estimate cannot be improved. Let $$s_k = ...
0
votes
2answers
46 views

Cauchy sequences are bounded?!

I'm having trouble understanding the proof that Cauchy sequences are bounded, here's the proof I've been given Let $s_n$ be a Cauchy sequence. We take a concrete value of $\varepsilon$, for ...
1
vote
3answers
34 views

Expand a sin(x^3) in Maclaurin's series and find a 30th derivative at (0)

I have a big task and problems with it. I have to expand this function in Maclaurin's series. $$cos(x^{3})$$ I tried expand it as $$\sum_{n=0}^\infty (x^n)/n!$$ but it's for $cos(x)$. So i don't know ...
0
votes
1answer
23 views

Absolute and conditional convergence of a series with $\sin(x)$

I have to explore absolute and conditional convergence of this function series I tried to find $a(n)$ and $a(n+1)$ terms of the series and then divide it and take a limit. But I've got nothing. ...
1
vote
0answers
18 views

Expand a function in Maclaurin's series

Please help me expand this function in Maclaurin's series to find an interval of convergence. I tried to turn it into this expression: but it doesn't give me the result. Help me please it's ...
3
votes
1answer
46 views

Interpretation of Riemann rearrangement theorem [closed]

There's a common thing that happens in mathematics, which is that all theorems are created equal, but some are more equal than others. Here are two examples of what I mean by that. (1) In Euclid, the ...
0
votes
1answer
20 views

Absolute and conditional convergence of function series

I have a problem. I have to explore absolute and conditional convergence of this function series Unfortunately. I didn't find in my reference any words about "absolute and conditional", Instead ...
1
vote
2answers
19 views

Calculate ch(0.2) to the nearest 0.01

Help me calculate ch(0.2) to the nearest 0.01. I tried to rewrite ch as a series but I still don't know how to evaluate it and what to do with factorial Help me please. it's very important
0
votes
3answers
48 views

Find a radius of convergence of power series

I have to Find a radius of convergence of this power series I' ve decided to use D'alambert indication: Looking for a limit i meet a problem with a factorial Please. help me finish this ...
1
vote
2answers
137 views

explore the convergence of series with ln(n)

Help me explore the convergence of red color rounded series. On this photo (the equation below) I used radical indication but it doesn't show me the result. What would be better to use? ...
0
votes
0answers
32 views

Convergence of a (easy) series

Let $ g:\mathbb{R}\to\mathbb{R}$ be defined as $g(x):=|x|$ if $x\in[-1,1]$ and extended with period $2$ on the whole real line. Prove that $\sum^{+\infty}_{n=0}(\frac{3}{4})^ng(4^nx)$ converges ...
5
votes
0answers
55 views

Forcing series convergence

The following problem was posed by one of my lecturers: $(z_n)$ a null sequence in $\mathbb{C}$. Does there exist $(\epsilon_n)$ with each $\epsilon_n=\pm 1$ such that: $$\sum_n \epsilon_n ...
0
votes
0answers
32 views

Question on Morse inequalities

I want to understand why: if i have then $(4.1)$ is formal : it means that please help me Thank you EDIT1: $(4.1)$ tel us that $\displaystyle\sum_{q=0}^{\infty} (M_q-\beta_q)t^q=(1+t)Q(t)$ ...
2
votes
2answers
39 views

Explore the convergence of a series

I have to explore the convergence of a series. At this picture I used radical Cauchy indication. But I don't now what to do with a denominator to find a limit. Help me please ! Thank You so much :) ...
0
votes
2answers
21 views

Find a sum of a series

Help me find a sum of this series I tried to excrete as (2/7)^n * 3^(n+2) and use De Lamber indication. It gives me a result 6/7. I checked it in Wolfram Math but the result was 54. Where did I go ...
2
votes
0answers
17 views

Convergence of Iteration with Sum

Consider a continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$, and the iteration $$ x_{k+1} = \frac{1}{k} \sum_{i=1}^{k} f( x_i ) $$ for some given initial condition $x_1 \in ...
1
vote
1answer
61 views

How to prove it?

Let $y_0\geqslant 2$, $y_n=y_{n-1}^2-2$, $n\in\mathbb{N}_+$, set $\displaystyle S_n=\sum_{k=0}^{n}\frac{1}{y_0\cdots y_k}$, how to prove $$\lim_{n\to\infty}S_n=\frac{y_0-\sqrt{y_0^2-4}}{2}.$$ Do you ...
0
votes
1answer
24 views

Is it possible to integrate this asymptotic expansion?

Suppose that some smooth function $f \in C^\infty\bigl(\mathbb R^n \times (0,+\infty)\bigr)$ possesses an asymptotic development $$ f(x,t) \sim t^{-\alpha} e^{ith(x)} \sum\limits_{k=0}^{+\infty} ...
0
votes
2answers
32 views

Explore the convergence of a series with ln

How to explore the convergence of this series: $$ \sum_{n=1}^{\infty}\dfrac{1}{\ln^n(n+1)} $$ What would be better to use: De Lamber indication or feature comparison. And if comparison is a good ...
0
votes
3answers
78 views

Determine whether this series converges or diverges:

I am taking a course in analysis and would just like to clarify my understanding with this motivating example? The series is: $$ \sum_{n=0}^\infty \frac{2^n - 1}{3^n + 1} $$ My textbook defines ...
0
votes
0answers
22 views

Prove the $d_\infty$ metric is finite

I need to show that the $d_\infty$ metric $$d_\infty(x,y) = \sup|x_i-y_i|$$ for all sequences $x$ such that $\sup|x_i| < \infty$ is indeed a metric by checking the metric axioms. I also need to ...