0
votes
0answers
37 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 ...
7
votes
2answers
135 views

Summing various rearrangements of $1-\frac12+\frac13-\frac14+\cdots$

Show that the series $1-\frac12+\frac13-\frac14+\ldots$ converges to $\log2$ but the rearranged series: ...
0
votes
3answers
44 views

Convergence of series $\sum$$u_n$= $\sum$$\frac{n! x^n}{(n+1)^n}$

My series is $$1+\frac{x}{2}+\frac{2! x^2}{3^2}+\frac{3!x^3}{4^3}+\ldots$$ My approach: $$u_n= \frac{n! x^n}{(n+1)^n}$$ So, $$u_{n+1}= \frac{(n+1)! x^{n+1}}{(n+2)^{n+1}}$$ So, ...
1
vote
1answer
50 views

Test for convergence of the series.

$$\frac{1}{1 \cdot 3}\ + \frac{2}{3 \cdot 5}\ + \frac{3}{5 \cdot 7}\ + \cdots$$ What is the $nth$ term here and what test should I use?
0
votes
3answers
43 views

Is this positive term series convergent?

My series is: $\frac{1}{1+2^{-1}}\ +\frac{1}{1+2^{-2}}\ +\ldots$ I see my $nth$ term is $\frac{1}{1+2^{-n}}$ How do I test for its convergence?
1
vote
1answer
23 views

Test for convergence of positive term series

$$\frac{2}{1^p}\ + \frac{3}{2^p}\ + \frac{4}{3^p}\ +\ldots\,.$$ I can see that the $nth$ term is $\frac{n+1}{n^p}$ How do I test for its convergence?
1
vote
1answer
22 views

Limit problem using sequential criteria for limits

$$\lim(n+n^2\log \frac{n}{n+1})= \frac12$$ How? In the text book it is simply said that this happens by Sequential criteria of limits. I don't get it.
2
votes
1answer
21 views

If $\{f_n\}$ and $\{g_n\}$ be uniformly convergent sequences of bounded functions on S, then $\{f_ng_n\}$ is uniformly convergent on S.

If $\{f_n\}$ converges uniformly to $f$ and $\{g_n\}$ converges uniformly to $g$, does it mean $\{f_ng_n\} $ will converge uniformly to $fg$? I am absolutely stuck on this. Please help.
0
votes
1answer
74 views

How to show the convergence of this infinite series: $\frac{x}{1+x}- \frac{x^2}{1+x^2}+ \frac{x^3}{1+x^3}\dots$

My series is $$ \frac{x}{1+x}-\frac{x^2}{1+x^2}+ \frac{x^3}{1+x^3}-..... $$ Given: $0<x<1$ I see that my nth term is $(-1)^{n+1} (\frac{x^n}{1+x^n})$ My approach was to use Dirichlet's test. ...
4
votes
1answer
50 views

Prove that the series is convergent: 1- (1+1/3)/2 + (1+1/3+1/5)/3-…

I can see that this is an alternating series with the $n$-th term $$(-1)^{n+1}\frac{1+\frac13+\frac15+\cdots+ \frac{1}{2n-1}}{n}.$$ What test can I apply to show that it converges? Also, it ...
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
42 views

Analysis Arithmetic series .Verify which of the following sequences converge.

Verify which of the following sequences converge.$$1.A(n)=\sum_{n=1,n=+00}(1/(n^{1+1/n})$$ $$2. B(n)=(1/\sqrt{n^2+1})+.......n/\sqrt{n^2+n}$$ $$3.C(n)=(n+cos(n^2))/(n+sin(n)) $$ .For the 3th one ...
0
votes
1answer
65 views

Does it converges? $ \sum \frac{\sqrt{n}\ln(n)}{n^2+1} $

As I checked on Wolfram Alpha I know that $$ \sum \frac{\sqrt{n}\ln(n)}{n^2+1} $$ Converges. But have tried many tests to show that, without success. I tried ratio/root (inconclusive). Cauchy test ...
0
votes
2answers
103 views

Does This Function Exist?

I am trying to construct a piecewise function $g:[0,1] \rightarrow \mathbb{R}$ with $g(0)=0,\hspace{3mm}$ $g \geq 0$, $\hspace{3mm}$ $\int_0^x g(t)dt\leq x,\hspace{3mm}$ and such that there is a ...
1
vote
1answer
25 views

Formula of Squaring Sums / Integrals

I'm trying to find a proof for the identities (which I use quite often) $$\left ( \int_{a}^{\infty}f(x)\,dx \right )^2=\int_{a}^{\infty}\int_{a}^{\infty}f(x, y)\,dx\,dy$$ and similarly for the series ...
1
vote
1answer
34 views

Can we expect to find $r,$ large enough, so, $\sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}\leq C (1+y^{2})^{s} $ for all $y\in \mathbb R$?

Fix $y\in \mathbb R$ and $s>1.$ Consider the series: $$I(y)=\sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}.$$ My Question is: Can we expect to find $r$ large enough, so that ...
2
votes
2answers
67 views

Can we expect to find some constant $C$; so that, $\sum_{n\in \mathbb Z} \frac{1}{1+(n-y)^{2}} <C$ for all $y\in \mathbb R;$?

Fix $y\in \mathbb R;$ and consider the series: $$\sum_{n\in \mathbb Z}\frac{1}{1+(n-y)^{2}}.$$ My Question is: Can we expect to find some constant $C$; so that, $$\sum_{n\in \mathbb Z} ...
1
vote
1answer
41 views

$\sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}\leq C$ for all $y\in \mathbb R$?

Fix $y\in \mathbb R$ and $s>1.$ Consider the series: $$I(y)=\sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}.$$ My Question is: Can we choose $r$ large enough so that $I(y)< C$ for ...
4
votes
4answers
82 views

a question how to prove:$\sum_{n=1}^{\infty}{{(-1)}^{n-1}{\cos(nx)}\over {n}}=\ln(2\cos(x/2))$

I found a complicated question in my textbook, I can't solve it? How to prove $$\sum_{n=1}^{\infty}{{(-1)}^{n-1}{\cos nx}\over {n}}=\ln(2\cos(x/2))$$ where $x\in(-\pi,\pi)$. My tried method: I tried ...
7
votes
1answer
184 views

Eigenvalues gone wild

I added some significant details to this problem, as it was apparently not clear to everyone what I want to know: This is a question about convergence of eigenvalues which essentially came up in ...
3
votes
1answer
99 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 ...
2
votes
1answer
46 views

Does the series $\sum_{i=0}^{\infty}\exp\{{-(m!)!}\}(D^m\phi)(0)$ converge for every $\phi \in C^\infty$?

Does the series $$\sum_{m=0}^{\infty}\exp\{{-(m!)!}\}(D^m\phi)(0)$$ converge for every $\phi \in C^\infty$? For analytic function $\phi$, we can show that the series converges by using ...
-1
votes
2answers
70 views

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

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 ...
2
votes
2answers
87 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 ...
2
votes
1answer
52 views

Series in a space which is not complete

Let $X$ be a normed vector space and $\left\lbrace x_n \right\rbrace_{n \in \mathbb{N}} \in X^{\mathbb{N}}$ with $$\sum_{n=1}^{\infty} \|x_n\| < \infty \wedge \sum_{n=1}^{\infty} x_n \notin X,$$ ...
1
vote
1answer
48 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?
0
votes
0answers
55 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 ...
2
votes
1answer
54 views

Generalizing the Monotone Subsequence theorem

In proving the Bolzaono-Weierstrass theorem, one proves the lemma that every infinite real sequence has a(n infinite) monotone subsequence. In all of the proofs I've seen so far, this is done by ...
1
vote
2answers
86 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. ...
2
votes
1answer
47 views

Series for reciprocal of polylogarithm?

The series representation of a polylogarithm of order $s$ is given by $$\text{Li}_s(z) = \sum_{k=1}^{\infty}\frac{z^k}{k^s}$$ Are there any simplified expressions for $\dfrac{1}{\text{Li}_s(z)}$? ...
1
vote
3answers
77 views

prove ${a_n}$ is a Cauchy sequence, provided $a_{n+2} = \frac{a_n + a_{n+1}}{2}$ [closed]

Suppose there is a sequence with the property $$a_{n+2} = \frac{a_n + a_{n+1}}{2}$$ for all $n \in \mathbb{N}_{+}$ Prove that ${a_n}$ is a Cauchy sequence I've self-taught ...
0
votes
3answers
61 views

On a certain series of complex numbers

Is it possible that the above infinite series is equal to ?
2
votes
3answers
94 views

Prove $\frac{\pi}{4} = \sum_{n = 0}^{\infty}\frac{(-1)^{n}}{2n + 1}$ using Dominated or Monotone Convergence

Is there a way to prove that $$\frac{\pi}{4} = \sum_{n = 0}^{\infty}\frac{(-1)^{n}}{2n + 1}$$ via the Dominated or Monotone Convergence Theorem?
3
votes
1answer
37 views

Prove the inequality based on an infinite series

Define $$f(x)=\sum_{n=1}^{\infty}\frac{nx^n}{1-x^n}.$$ It is easy to see that this series converges for $x\in(-1,1).$ Now we are asked to show that $(1-x)^2f(x)\geq x,$ for $x\in[0,1).$ I tried ...
4
votes
1answer
82 views

Prove the inequality for all $N$

Show that the following inequality holds for all integers $N\geq 1$ $$\left|\sum_{n=1}^N\frac{1}{\sqrt{n}}-2\sqrt{N}-c_1\right|\leq\frac{c_2}{\sqrt{N}}$$ where $c_1,c_2$ are some constants. I have ...
-2
votes
1answer
26 views

Sum of a convergent and divergent sequence [closed]

If $(x_n)$ is a convergent sequence and $(y_n)$ is a divergent sequence, show that their sum diverges.
1
vote
2answers
44 views

uniform convergence of a functional sequence

Is this sequence of functions $$f_n(x)=n^3x(1-x)^n$$ converges uniformly for $x\in[0,1]$. I need some help on this.
8
votes
4answers
617 views

Can every positive real be written as the sum of a subsequence of dot dot dot

I answered this thing Infinite sum of prime reciprocals and now wonder what happens if we do not have such a strong condition as Bertrand's postulate. i have been fiddling with this, not sure either ...
0
votes
1answer
63 views

Convergent's numerators of the continued fraction for $\pi$

Call $C_{\pi} = \{ 1,3,22,333,355,… \}$, it´s the sequence of the numerators of convergents of the continued fraction for $\pi$, its OEIS' A002485, http://oeis.org/A002485. Let $n \in C_{\pi}$, such ...
1
vote
1answer
95 views

A sum for stirling numbers Pi, e.

In this identity $$1-e{}^{2} = \displaystyle \sum _{n=0}^{\infty } \frac{(-1)^n(\pi )^{2 n}} {(2 n)!}\sum _{k=0}^{2 n} (-1)^{k} S_2(2 n,1-k+2 n),$$ $S_2$ is a Stirling number of the second kind. ...
1
vote
1answer
36 views

Laurent series of $\frac{e^{iz}}{z^2+p^2}$, $ p>0$.

I need help finding the main part of the laurent series of $f(z)=\frac{e^{iz}}{z^2+p^2}$ in $ip,-ip$ since these are the two poles of $f$. Due to the orders of the poles are 1 I just have to find ...
2
votes
1answer
62 views

the series $\sum_{k=1}^\infty a_k$ converges implies the series $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational

Let $\sum_{k=1}^\infty a_k$ be a convergent series. Then can we obtain $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational? If $\sum_{k=1}^\infty a_k$ converges absolutely, then I can ...
-1
votes
4answers
145 views

Does $\sum_{j = 1}^{\infty} \sqrt{\frac{j!}{j^j}}$ converge? [closed]

I need to solve $$\sum_{j = 1}^{\infty} \sqrt{\frac{j!}{j^j}}$$ Does this converge or diverge and why?
3
votes
1answer
93 views

How Find the sum $\sum_{n=2}^{\infty}\left(\sum_{j=1}^{n}\frac{1}{(2j-1)^2} \binom{2n}{n}\frac{1}{2^{2n}(2n+1)}\right)$

Prove or disprove $$I=\sum_{n=2}^{\infty}\left(1+\dfrac{1}{3^2}+\dfrac{1}{5^2}+\cdots+\dfrac{1}{(2n-1)^2}\right) \binom{2n}{n}\dfrac{1}{2^{2n}(2n+1)}=\dfrac{\pi^3}{48}-\dfrac{1}{6}$$ My ...
5
votes
2answers
371 views

Series of sequence converges?

Given the recursively defined sequence $$ a_2 = 2(C+1)a_0 $$ and $$ a_{n+2}=\frac{\left(n(n+6)+4(C+1)\right)a_n - 8(n-2)a_{n-2}}{(n+1)(n+2)}. $$ that I got from the Frobenius method applied to an ODE, ...
0
votes
0answers
27 views

geometric convergence of sequence

Let $\{x_n\}$ be a sequence of non-negative real numbers. For fixed $m \in \mathbb{N}$ Denote $$x_k^{(m)}:={1 \over m}\sum_{i=1}^m x_{k+1-i},\forall k\geq m$$ Assume $x^{(m)}_k\leq ...
0
votes
1answer
35 views

if $f(x)=\sum_{n=1}^{\infty}\frac{x^n}{n^2\ln{(n+1)}}$,show that $f$ is differentiable on $x=-1$ and $x=1?$..

let $$f(x)=\sum_{n=1}^{\infty}\dfrac{x^n}{n^2\ln{(n+1)}},-1\le x\le 1$$ prove $f$ is differentiable on $x=-1$ and $x=1?$.. for this problem I want show $f(x)$ is uniform convergence? maybe ...
0
votes
2answers
34 views

Pointwise and uniform convergence of this series

$$\sum_{n=1}^{\infty}\left(1- \frac{1}{2n}\right)^{-n^2}(x^2-1)^n$$ I've tried treating it as a power series centered around $x = 1$ and $x = -1$ and using root test I arrive to radius of convergence ...