0
votes
0answers
13 views

Convergence of Series of Complex Numbers with Decreasing Modulo (non-zero imaginary part)

Let $(a_n)_{n \in \mathbb{N}}$ be a decreasing sequence of positive real numbers tending to zero. Show that for $\theta \in \mathbb{R}$, $\theta$ not a multiple of $2\pi$, the series $\sum_{n\geq1} ...
1
vote
1answer
22 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 ...
5
votes
0answers
57 views

Convergence of infinite series of complex numbers [duplicate]

This has been bugging me for some months since our lecturer, a fields medalist, mentioned that he couldn't solve it when he was our age, yet had had two students submit solutions to it (during our ...
0
votes
1answer
7 views

Sequence Interval of convergence

I could someone help me with the following sequence of functions of which I attempted to find the interval of convergence, but I couldn't get it to match with the solution I get from WolframAlpha ...
2
votes
1answer
24 views

Find the values of $p$ for which the given sequence converges in $l^p$ norm or weakly

We consider $E_p=(c_{00}, ||\cdot||_{p})$ (where $c_{00}$ are the sequences that are zero except in a finite number of values and $1\leq p \leq \infty$) and the sequence:$$(x_n)_{n\in ...
0
votes
2answers
40 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 $f(a+h)$?
3
votes
1answer
65 views

If a recursive sequence converges, must its inverse be divergent?

Suppose I have a recursive sequence $\displaystyle a_{n+1} = \frac{a_{n}}{2}$. Clearly, the sequence converges towards zero. Now, suppose I define an "inverse" sequence $\displaystyle b_{n+1} = ...
1
vote
2answers
54 views

How to prove or disprove the convergence of the series $\sum_{n=1}^{\infty}\frac{(-1)^n\cdot a_{n}}{n}$?

Question: Assume that $a_{n}\in\mathbb R$, and let the series $$\sum_{n=1}^{\infty}a_{n}$$ be convergent I would like to prove or disprove the convergence of ...
3
votes
2answers
68 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
62 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
1answer
48 views

$\lim_{x\rightarrow {0}^{+}}\sum_{n=1}^{\infty}{(-1)}^{n-1}\frac{1}{n!{x}^{n}}=?$

$$\lim_{x\rightarrow {0}^{+}}\sum_{n=1}^{\infty}{(-1)}^{n-1}\frac{1}{n!{x}^{n}}=?$$ I found this question during my study. In my opinion ,it is not difficult to solve ,but it is interesting. So I ...
4
votes
5answers
172 views

Is the series $\sum _{n=1}^{\infty } (-1)^n / {n^2}$ convergent or absolutely convergent?

Is this series convergent or absolutely convergent? $$\sum _{n=1}^{\infty }\:(-1)^n \frac {1} {n^2}$$ Attempt: I got this using Ratio Test: $$\lim_{n \to \infty} \frac{n^2}{(n+1)^2}$$
0
votes
1answer
31 views

$\Delta^kn^\alpha$ converges monotonically to zero when $\alpha<k$

I am trying to prove that the $k$-th finite difference of the series $n^\alpha$ converges to zero monotonically as $n\to\infty$ when $\alpha<k$. The differential analogue is ...
1
vote
1answer
183 views

How to find a series for comparison with $\sum 1/\sqrt{n(n+1)}$?

The series $$\sum_{n=1}^{+\infty} \frac{1}{\sqrt{n(n+1)}}$$ I have tried ratio and integral both lead me to inconclusive, so probably it's by comparisson but I can't find What to compare.
3
votes
2answers
68 views

$\lim \sqrt[n]{a^n + b^n}$

I've seen some answers here for why this limit is the maximum between $a$ and $b$, but all of then included the hypotesis that $a$ and $b$ are both non negative. It was asked to show that this limix ...
2
votes
2answers
59 views

Convergence for all $\theta$ of a sum with periodic function

How can I show that: $$ \sum_{n \geq 1} \dfrac{\sin(n\theta)}{n} $$ converges for all $\theta \in \mathbb{R}$?
0
votes
2answers
52 views

convergence of sin functions absolutely for all x in $\mathbb R$ [closed]

How do I show that $\sum_{n = 1}^{\infty}2^n \sin(x/3^n)$ converges absolutely for all $x \in \mathbb{R}\;$?
3
votes
2answers
65 views

Radius of convergence of power series which has factorial term

I am trying to find radius of convergence of the following power series: $\sum_{n\geq 1} n^n z^{n!}$ I tried ratio test but it became complicated, I have never seen such radius of convergence problem ...
1
vote
2answers
31 views

Divergence of a recursive sequence

If $(x_n)$ isthe sequence defined by $x_1=\frac{1}{2}$ and $x_{n+1}=\sqrt{x_n^2 +x_n +1}$, show that $\lim x_n = \infty$ Ive tried a couple of things but none of them helped. Ive tried to suppose, by ...
0
votes
1answer
27 views

Sequence of sets $f_n$ that converge almost everywhere to $f$ but not almost uniformly to $f$?

I've been trying to think of a simple example of a sequence of sets $f_n$ that converge almost everywhere to $f$ but not almost uniformly to $f$. Suppose $f$ is the zero function for the sake of ...
0
votes
2answers
37 views

How to use the Comparison Test to investigate the convergence of $\sum (\ln n)/n^\alpha$?

Let $$\sum\limits_{n=1}^\infty \frac{\ln n}{n^\alpha}, \alpha\in\Bbb{R}$$ I need to investigate the convergence of this series. I've read that since the series is positive for all $n$ then it ...
2
votes
2answers
54 views

Verifaction of convergence/divergence exercise

I have the following assignment in my textbok: Series $\sum_{n=0}^{\infty}c_{n}3^n$ is convergent. Based on that can we conclude that the following series coverge: a) $\sum_{n=0}^{\infty}c_{n}2^n$ ...
0
votes
1answer
49 views

Sequence in $\mathbb R^2$ converges if and only if it is Cauchy

How to prove that a sequence $(x_n)$ in $\mathbb R^2$ converges if and only if it is Cauchy? I've proven that the triangle inequality holds for the euclidean norm of vectors.
2
votes
2answers
59 views

Find the first 5 coefficients of the series $\frac{6x}{x+9} = \sum_{n=0}^\infty C_n x^n$

I rewrote the equation series as $$ \frac69 \sum_{n=0}^\infty \left(\frac{-1}{9}\right)^n x^{n+1} $$ And therefore have coefficients of $C_0 = 6/9, C_1 = \left( 6/9 \right) ...
-4
votes
2answers
97 views

What does this infinite sum converge to?: $\sum_{n=1}^\infty \dfrac{1}{n^k} = \dfrac{1}{1^k} + \dfrac{1}{2^k} + \dfrac{1}{3^k} + …$

$$\sum_{n=1}^\infty \dfrac{1}{n^k} = \dfrac{1}{1^k} + \dfrac{1}{2^k} + \dfrac{1}{3^k} + \dfrac{1}{4^k} + \dfrac{1}{5^k} + ...$$ I've found that: when $k=1$, it diverge to infinity when $k=2$, it ...
4
votes
2answers
87 views

Convergence of tetration sequence.

This question arose from here. I am interested to find a nice proof about the convergence of $${^n}a=\underbrace{a^{a^{\ .^{\ .^{\ .^a}}}}}_{n\ \text{times}}.$$ I find with google a necessary and ...
4
votes
2answers
107 views

$\lim_{x\rightarrow 1}\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^x}=\ln2$.

Prove $$\lim_{x\rightarrow 1}\sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{{n}^{x}}=\ln2.$$ Of course $$\sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{{n}}=\ln2,$$ but we can not use the Proposition : If a ...
3
votes
1answer
82 views

Compute a sequence $A_n$ such that $\sum\limits_{n=1}^{\infty}\frac{1}{A_n\ln(A_n)}=1$

How can we compute a sequence $A_n$ of positive real numbers, such that $\displaystyle\sum\limits_{n=1}^{\infty}\frac{1}{A_n\ln(A_n)}=1$? One way I can think of, is by defining $A_n\ln(A_n)=2^n$, but ...
0
votes
0answers
37 views

Why there's no articles about the eta function convergence?

I've been searching about a proof that the eta function converges for $\mbox{Re}(z)>0$ but the ONLY page I've found that claims to prove it was in this question: ...
0
votes
2answers
52 views

Finding the convergence radius of $\dfrac{(n!)^k\cdot x^n}{(kn)!}$

If K is a integer positive, find the convergence radius of the series $$\sum\limits_{n=0}^{\infty} \dfrac{(n!)^k\cdot x^n}{(kn)!}$$ Any initial idea?
1
vote
1answer
25 views

Product of `statistically convergent' sequences

Let $F$ be the filter of statistical convergence on $\mathbb{N}$ (see Definition 2.1 here). Suppose that $(\xi_n), (\eta_n)$ are sequences of real numbers that converge to $\xi$ and $\eta$ along $F$, ...
1
vote
2answers
59 views

How to prove that the sequence is decreasing $a_{n}=\frac{ln(n)}{n^2}$

Is my way/proof good and completely mathematically rigorous? $a_{n}=\frac{ln(n)}{n^2}$ --> $a_{n+1}=\frac{ln(n+1)}{(n+1)^2}$ $\frac{ln(n)}{n^2} > \frac{ln(n+1)}{(n+1)^2}$ ...
3
votes
5answers
133 views

Convergent or divergent $\sum_{n=1}^{\infty}\frac{1}{\sqrt{n^2+1}}\left(\frac{n}{n+1}\right)^n$?

Any suggestions? I have tried using D'Alembert's test, but on the end I get 1. I can't think of any other series with which to compare it. In my textbook the give the following solution which I don't ...
1
vote
3answers
111 views

Decide convergence divergence of $\sum \dfrac{1}{(\ln n)^{\ln n}}$ [duplicate]

Does the series $\sum \dfrac{1}{(\ln n)^{\ln n}}$ converges? I can intuitively say that it converges, because $(\ln n)^{\ln n} $ is going to $\infty$ on a hayabusa
0
votes
2answers
61 views

Find a sum of $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{ch(n)}{3^n}$

Find a sum of $$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{ch(n)}{3^n}$$ Could you give some some hint or some way to start this? I have tried representing ch(n) through its definition with e, but I ...
3
votes
3answers
105 views

Does $\sum_{n=1}^{\infty}\frac{n-1}{n^2}$ converge or diverge?

Is my logic OK? $a_{n}=\frac{n-1}{n^2}$ $\frac{1}{n} \leq b_{n}=\frac{n-\frac{n}{2}}{n^2}=\frac{n}{2n^2}=\frac{1}{2n} \leq a_{n}=\frac{n-1}{n^2}$ and there for the initial series diverges.
3
votes
1answer
32 views

Prove that $(x_n)_{n\geq1}$ is an arithmetic progression

Let $(x_n)_{n\geq1}$ be a sequence of integers. Define $y_n=\frac{x_n}{n},n\geq1$. The sequence $(y_n)_{n\geq1}$ is convergent and $n$ divides the sum of any $n$ consecutive terms of the sequence ...
1
vote
0answers
59 views

Does this convergence test for series hold?

Long ago before I joined math.se, a friend asked this question here after he and I had discussed it some. An answer was accepted that is narrow in scope, so I am going to ask a more specific version ...
3
votes
1answer
26 views

Why does $\lim\limits_{N \rightarrow \infty}{\sum_{i=1}^{N}\frac{1}{\frac{N}{1-\epsilon}-i}}$ converge to $\log\left[\frac{1}{\epsilon}\right]$?

while playing around with my equations, i found that the following has to hold for my universe to be consistent: $$\lim_{N \rightarrow ...
0
votes
1answer
23 views

Convergence of a Cauchy sequence of matrices

I have a Cauchy sequence of matrices $C_i \in R^{p \times q}$, i.e. $\lim_{n\rightarrow \infty} \| C_{n+1} - C_{n} \| = 0$ for any norm (I just need the property that $\|C_1-C_2\|>\delta ...
3
votes
2answers
60 views

Finding the convergence

The sequence $a_n = \sqrt[n]{(3^n+5^n)}$ is convergent? Tried to resolve applying the the limit $n\to\infty$ but couldn't figure out how to finish it, any idea? Thanks
1
vote
4answers
61 views

Why does $\sum_{k=1}^{\infty} \frac {(-1)^{k-1}}{k}$ converge conditionally?

Why does it converge conditionally? $$\sum_{k=1}^{\infty} \frac {(-1)^{k-1}}{k}$$
2
votes
4answers
53 views

How to apply the alternating series test to the series $\sum (-1)^{n+1} n/2^n$?

So, I need to test the following series for convergence or divergence: $$\sum_{n=1}^\infty (-1)^{n+1}{n\over {2^n}}$$ I know that when you use the Alternating Series Test, the series must satisfy ...
1
vote
3answers
48 views

Is the following series converging or diverging. $\sum_{n=1}^{\infty}\dfrac{n+4^n}{n+6^n}$

I know one solution. That is by Doing comparison with $\dfrac{4^n+4^n}{6^n}$ Wondering if there are more ways to do it
0
votes
2answers
39 views

Problem with proof that positive infinite series are commuative

Proof from real analysis book: Let $\sum_{n=0}^{\infty}a_n$ converge, where $a_n \geq 0, n \in \mathbb{N}$. Then the series $$ \sum_{k=1}^{\infty}a'_k = a'_1 + a'_2 + \cdots + a'_k + \cdots $$ ...
2
votes
4answers
70 views

Convergence of $a_n=1+1/5+1/9+\ldots+\frac{1}{4n-3}$

Show that the sequence $$a_n=1+1/5+1/9+\ldots+\frac{1}{4n-3}$$ does not converge but the sequence $b_n=\frac{a_n}{n}$ converges. I can show the first part. For the second part, will it be sufficient ...
4
votes
0answers
40 views

How to integrate scalar field over quarter torus? Infinite series does not converge.

This seems to be physics question, but the problem just concerns math. Preface If one wants to calculate the permeance $P$ of a rectangular bar: it is an easy task: $$P = \frac{\mu a b}{L} ...
0
votes
1answer
35 views

Simpler way of proving series convergence?

Determine whether the following series converges $$\sum_{n=1}^{\infty} \left (\frac{n^4}{n^4 + 2}\right)^{n^5-3}.$$ I've found convergence using the root criterion in the following way. $\sqrt[n]{ ...
1
vote
3answers
68 views

A sequence and convergence

Let $\{x_n\}$ and $\{y_n\}$ be sequences of real numbers which converge to $\ell$ and $m$ respectively. Show that $$\lim_{n\to \infty} \frac{1}{n}\sum_{k=1}^{n}x_ky_{n+1-k}=\ell m$$ This is a ...
3
votes
0answers
26 views

Convergence of sum of antiderivative and derivative

This question is inspired by this question: Solutions for $ \frac{dy}{dx}=y $?. It makes me wonder if there are any function where the sum of all antiderivative and derivative converges. The ...