Questions tagged [conditional-convergence]

This tag is for questions related to conditional convergence. A series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.

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32 views

Conditional convergence for improper Riemann double integrals

I'm reading Buck's advanced calculus. It says for improper integral of higher dimensions, conditional convergence is impossible, i.e., $\int\int_D f$ cannot exist without $\int\int_D|f|$ existing too. ...
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27 views

Convergence of Euler product implies convergence of Dirichlet series?

Suppose we have an Euler product over the primes $$F(s) = \prod_{p} \left( 1 + \frac{a_p}{p^s} \right)^{-1},$$ where each $a_p \in \mathbb{C}$. The Euler product is convergent in the range $Re(s) > ...
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71 views

If $\sum a_n^k$ converges for all $k \geq 1$, does $\prod (1 + a_n)$ converge?

By definition, an infinite product $\prod (1 + a_n)$ converges iff the sum $\sum \log(1 + a_n)$ converges, enabling us to use various convergence tests for infinite sums, and the Taylor expansion $$ \...
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26 views

Convergence of a rearrangement of conditionally convergent series

$\{a_n\}$ is a sequence of real numbers.$\space\sum_{n=1}^{\infty} a_{2n}$ and $\sum_{n=1}^{\infty} a_{2n-1}$ are both conditionally convergent. Is there such $\sum_{n=1}^{\infty} a_{n}$ that is ...
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44 views

What is the domain of convergence for the Taylor Series for this function?

What is the domain of convergence in variable $a$ of the Taylor Series of this function? $$h\left(a\right)\equiv-\int_{-\infty}^{\infty}p\left(x,a\right)ln\left(p\left(x,a\right)\right)dx$$ ...
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59 views

Which sequence ${a_n}$ does $\sum_{n=1}^\infty a_n$ is conditionally convergent and $\sum_{n=1}^\infty (-1)^n a_n$ converges

Which sequence ${a_n}$ does $\sum_{n=1}^\infty a_n$ is conditionally convergent and $\sum_{n=1}^\infty (-1)^n a_n$ converges? I tried with ${a_n}= \frac{\sin(n)}{n}$ and it seems that it does. But I ...
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29 views

Are the partial sums of $\sin (n)$ bounded? [duplicate]

My Calculus II teacher left us the task of finding a sequence that is conditionally convergent, and that if we multiply it by $(-1)^n$ it converges. I think that sequence could be $\frac{sen(n)}{n}$. ...
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34 views

Rearrange conditionnaly convergent sequence so the partial sums form a dense subset of $\mathbb{R}$

I was reading about conditional convergence vs absolute convergence. I understand from the wikipedia on Riemann series theorem how we construct a permutation such that the partial sum converges to any ...
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39 views

Reorganising the order of a conditionnaly convergent series.

I was reading this article on wikipedia about reorganizing the order of summation of a series. I dont understand why if we reorganise the series $0=1-1+\frac{1}{2}-\frac{1}{2}+\frac{1}{3}-\frac{1}{3}+...
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1answer
43 views

How the series is convergent and absolute convergent

$$\sum_{n=1}^\infty (-1)^{n-1} \frac{\sin(nx) + \cos(nx)}{n^{3/2}}$$ I tried to prove convergence by Leibnitz theorem but can't prove absolute convergent please give me some hint about this.
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68 views

Sum of series with conditional convergence

Sorry for this question, but for some reason I'm stuck on this for few hours already. Before I solved more complex ( I think ) problems, but can't solve this. The only thing I know that this series ...
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68 views

For which $x$ is $ \sum_{n=1}^{\infty} \frac{(x-b)^{n}}{na^{n}}$ (absolutely) convergent?

Consider the power series: $$ \sum_{n=1}^{\infty} \frac{(x-b)^{n}}{na^{n}}$$ with a, b >0 : a) for which x is this series absolute convergent, b) for which x is this series conditionally convergent, ...
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82 views

Why does the derivative of Dirichlet integral $\int_0^\infty \frac{\sin{ax}}{x} dx$ with respect to $a$ gives nonsense?

If I define $I(a)=\int_0^\infty \frac{\sin{ax}}{x} dx$. We know that $\int_0^\infty \frac{\sin{ax}}{x} dx=\frac{\pi}{2}$. If I differentiate both sides by $a$ then we get $\int_0^\infty \cos{ax} dx=0$....
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60 views

proving if $\sum_{n=2}^{\infty} \frac{n\cdot \cos(2n)}{n^2 + 11} + \frac{\cos(\pi n)}{\ln(n)\cdot \ln(n^n + n)}$ converge conditionally or absolutely

I have already proven that $$\sum_{n=2}^{\infty}\frac{\cos(\pi n)}{\ln(n)\cdot \ln(n^n + n)}$$ conditionally converge and $$\sum_{n=2}^{\infty} \frac{n\cdot \cos(2n)}{n^2 + 11}$$ converge, hence $$\...
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45 views

Does the ordering of a Schauder basis matter in Hilbert space?

If $S=\{v_i\}_{i\in\mathbb N}$ is a (not necessarily orthogonal) Schauder basis for a Hilbert space $H$, must $S$ be an unconditional Schauder basis? I define these terms below because not every ...
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1answer
58 views

What can I say about the terms of the sequence of partial sums $\{S_k \}$ of the conditionally convergent series $\sum\limits_{n=1}^{\infty} a_n$?

Suppose $\sum\limits_{n=1}^{\infty} a_n$ be a conditionally convergent series with $\sum\limits_{n=1}^{\infty} a_n = 0$ and the sequence of partial sums $\{S_k \}.$ Can it happen that $S_k > 0$ ...
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1answer
29 views

Convergence or divergence of when the alternating series test fails.

I want to investigate the convergence or divergence of the following series:$$\sum _{n=0}^{\infty }\:\frac{\left(n!\right)^2}{\left(2n\right)!}(-4)^n$$ However, the alternating series test fails ...
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1answer
31 views

A series conditionally convergent to 0 whose sequence of partial sums is positive.

Does there exist a series conditionally convergent to 0 with sequence of partial sums {sn} such that sn is positive for every n? I considered 1-1+1/2 -1/2 +1/3 -1/3 + ....This series converges ...
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1answer
26 views

Finding the values of x for which this series converges

The questions asks... $$\textrm{For what values of }x \textrm{ do the following series converge, where }a > 0,\, b > 0?$$ I have managed to find the correct values of $x$ for the first two ...
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21 views

boundedness of elements of a converging series

The background of my question is that I am trying to study properties of neural networks. In this analysis, I ran into the following question: Consider the following function $$ f_n(x) = \frac{1}...
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22 views

Let $\sum a_n=s$ be conditional convergent, $\sum a_{f(n)}=t\neq s$. Show $\forall\ N, \exists\ n$, such that $|n-f(n)|>N$.

Let $\sum a_n=s$ be conditional convergent, $\sum a_{f(n)}=t\neq s$. Show $\forall\ N, \exists\ n$, such that $|n-f(n)|>N$. Here, $f: \Bbb N\to\Bbb N$ is a bijection. How to prove? My attempt. ...
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45 views

Let $f(x)$ be the $2π$-periodic function defined by $ f(x)= \begin{cases} 1+x&\,x \in \mathbb [0,π)\\ -x-2&\, x \in \mathbb [-π,0)\\ \end{cases} $

Let $f(x)$ be the $2π$-periodic function defined by $ f(x)= \begin{cases} 1+x&\,x \in \mathbb [0,π)\\ -x-2&\, x \in \mathbb [-π,0)\\ \end{cases} $ Then the Fourier series of $f$ $(A)$ ...
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1answer
196 views

Does $\iint_D \frac{x^2}{x^2+y^2} dx dy $ converge on $D= \left\{ (x, y) : x^2+y^2\leq ax \right\} $ ? If yes, what value does it converge to?

Powers equal to $2$ entice the polar substitution. $D= \left\{ (x, y) : x^2+y^2\leq ax \right\}$, so $0 \leq r \leq a \cos \phi.$ For the domain to make sense, we need either $\phi \in [0, \frac{\pi}{...
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37 views

How do I need to use Euler-Maclaurin summation formula to calcuate the limits $\lim S_n$?

Dear Sirs and Dear Madams, I need to calculate the limit of $$S_{n} = \sum\limits_{k =1}^{n} [ \dfrac{1}{4k-3} + \dfrac{1}{4k - 1} - \dfrac{1}{2k} ] .$$ I show that \begin{eqnarray*} \lim\limits_{n \...
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30 views

Definition of convergence in conditional distribution

It seems obvious to me that we can generalize the concept of weak convergence of probability measures to regular conditional probability measures. The standard definition for a sequence of measures $\...
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184 views

Rearrangement in proof for Euler's formula

In the proof for Euler's formula, we expand $e^{ix}$ as a Taylor series, rearrange the terms, factor out $i$, and thus obtain the Taylor series for $\sin (x)$ and $\cos(x)$. However, this ...
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1answer
27 views

What is the limit of the nth root of a sequenc which diverges?

The question under power serieses of complex numbrs. The main question is: Prove that if a given sequence converges conditionally. Then, the converging radius of The power series of that sequence ...
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70 views

Is $\sum{\frac{i^{n}}{n}}$ convergent or divergent?

Consider the series $$\sum{\frac{i^{n}}{n}}$$ We know that this series is not absolutely convergent as $\sum{|z_{n}|}$ gives harmonic series which is divergent. However this series could be ...
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1answer
38 views

How to show a series is conditionally convergent

$a_k=\begin{cases}\frac{1}{(k-1)^2} - \frac{1}{k}, n \text{ is odd }\\ \frac{1}{k-1} - \frac{1}{k^2}, k \text{ is even }\end{cases}$ I have to show the series is conditionally convergent, i.e; the ...
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1answer
31 views

The series for absolute and conditional convergence [closed]

$$\sum_{k=1}^{\infty} \frac{\cos(\frac{\pi k}{3})}{2^{\ln(k)}} $$
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28 views

Explore the series for absolute and conditional convergence

$$\sum_{k=1}^{\infty} \frac{(-1)^{k} k}{\sqrt{(k+1)(k+2)}} $$ I found that absolute series diverges
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68 views

A detailed proof of Riemann rearrangement theorem

I'm trying to formulate the ideas from here and here into a rigorous argument. Honestly, this attempt takes me the entire three days to complete, and it is full of indexes of subscripts and ...
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1answer
39 views

Convergence of integral: $ \int_{0}^{\infty} \frac{\sin{\left(x+\frac{1}{x}\right)}}{x^{a}}\ dx $

I need to determine about convergence of integral below.I found out that it doesn't converge absolutely but i have problem with determining about conditional convergence.I am not sure how to use ...
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48 views

Riemann rearrangement theorem: Existence of a rearrangement that sums to any real number $M$

I'm trying to fill in the detail for the proof of Riemann rearrangement theorem from this Wikipedia's page. Theorem: If $\sum x_{k}$ is a conditionally convergent series in $\mathbb{R}$, then, ...
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1answer
55 views

If $\sum x_k$ is converges conditionally, then $\sum x^+_k$ diverges

I'm working on this problem If $\sum x_k$ is a conditionally convergent series in $\mathbb R$, then $\sum x^+_k$ diverges where $x^+_k = \max \{x_k,0\}$. Could you please verify if my attempt is ...
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54 views

almost sure convergence conditional on the sample

I have the following problem (any help is really appreciated). Given a sample of size $N$, I estimate a quantity (it does not matter what, but for the records it is a scalar, fixed coefficient), and ...
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1answer
621 views

Can $e^x$ be expressed as a linear combination of $(1 + \frac x n)^n$?

Can $e^x$ be expressed as a linear combination of $(1 + \frac x n)^n$? In other words, does there exist an infinite sequence $(a_k)_{k \in \mathbb N_0}$ such that $$e^x = a_0 + \sum_{1 \leq k < \...
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1answer
111 views

Improper Lebesgue integral $\int_{\mathbb R}\frac{\sin(x)}x~\mathrm dx$

While thinking over the idea of an improper Lebesgue integral, I came up with the following: $$\int_0^\infty\mu\{x~|~f(x)>t\}-\mu\{x~|~f(x)<-t\}~\mathrm dt$$ with $\mu$ being the Lebesgue ...
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55 views

Is square of Poisson kernel conditionally convergent?

Poisson kernel is defined for $\theta\in [0,2\pi)$ as $$P_r(\theta) = \sum\limits_{n\in\mathbb{Z}} r^{|n|}e^{in\theta}$$ which equals $$P_r(\theta) = \frac{1-r^2}{r^2-2r\cos (\theta) + 1 }.$$ For this ...
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3answers
52 views

Convergence of the series below

$$\sum_{n=1}^\infty\frac{(-1)^n}{\sqrt{n}}$$ I did: $$\lim_{n\to \infty}\Biggr\vert\frac{(-1)^n}{\sqrt{n}}\Biggr\vert$$ $$\lim_{n\to \infty}\frac{1}{n^\frac{1}{2}}=0<1$$ So diverges by the Ratio ...
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53 views

The power series $\sum \frac{(x-b)^n}{na^n}$ with a,b>0?

The power series $\sum \frac{(x-b)^n}{na^n}$ with a,b>0 ? How do i show for which x the series is conditionally convergent? Do i have to express in terms of a and b. or something like that?
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1answer
32 views

Could I get an explanation on why this would conditionally converge?

$$\sum\limits_{n=2}^\infty \frac{\cos(n\pi)}{\ln(n)^2}$$ I'm not sure how this would conditionally converge, according to my calculations I would assume it's absolutely converge.
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33 views

Numerical Solver Exercise

Sensitivity to initial conditions is well illustrated by a little target practice with your numerical solver. Enter $x' = x^2 - t$ into your numerical solver, and then experiment with initial ...
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35 views

Is $\sum_{n=1}^{\infty} \frac{(-1)^n(3n^3+4n^2)}{4+2n^5}$ divergent, conditionally convergent, or absolutely convergent?

The first thing I did was create a sequence ($A$) for what was inside of the sum, then I created another sequence ($B$) that is related to $A$. $$A=\frac{(-1)^n(3n^3+4n^2)}{4+2n^5}$$ $$B=\frac{1}{n^2}...
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2answers
119 views

Show that if $\sum a_n$ converges absolutely, then $\sum (\tfrac{n+1}{n})a_n$ also converges absolutely.

Prove that if $\sum_{n=1}^\infty a_n$ is convergent, then the series $$\sum_{n=1}^\infty \left(\frac{n+1}{n}\right)a_n$$ is also convergent. Edit: Originally, I thought about doing a limit comparison ...
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1answer
64 views

Conditional convergence of $\sum_{n=1}^{\infty}\frac{\sin n}{n^p}$

I am working on the exercise of determining when $\sum_{n=1}^{\infty}\frac{\sin n}{n^p}$ converges conditionally. I know if $p\leq0$ it cannot converge by the vanishing condition, and if $p>1$ the ...
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0answers
33 views

Rearranging Terms of Conditionally Convergent Series

I know about Riemann’s Rearrangement Theorem. But certainly you can rearrange some of the terms of a conditional series and still arrive at the same series value. For example, if you only rearrange a ...
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2answers
130 views

Working out which x for the Conditional Convergence of a Series

I'v got roughly half way through this question: For (fixed) x which is an element of the real numbers, consider the series $\sum_{n=1}^\infty \frac{x^{n-1}}{2^nn} $ For which x does this series ...
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3answers
56 views

Why is $\sum_{k=2}^{\infty} \frac{(-1)^k}{k\ln k}$ conditionally convergent?

I was solving this question, and I thought it would be absolutely convergent because when you set the modulus, you would get $$\sum_{k=2}^{\infty} \frac{1}{k\ln k}.$$ But the answer says it is ...
3
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1answer
39 views

critical case in $L^p$ convergence

Let $f_n$ be a bounded sequence in $L^1\cap L^{p}$ with $1<p<\infty$. Assume that $f_n\to f$ strongly in $L^1$. Then basically we have $f_n\to f$ strongly in $L^q$ for all $1\leq q<p$. But we ...