Convergence of sequences and different modes of convergence.

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Why do we assign values to divergent series?

Why do we assign values to divergent series? For example, the series $1+2+3+4... = -1/12$. I understand the proof for this, but I feel like it uses false math, and I recall reading that you can't do ...
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1answer
10 views

Range of convergence for Taylor's series for e^(sin x)

Is there anything wrong with my method below? Also, is there an easier method? For $sin\,x = \sum^{\infty}_{k=0}\frac{(-1)^kx^{1+2k}}{(1+2k)!}$, $L_1 = \lim_{k\rightarrow\infty} \left| ...
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1answer
57 views

Does the series $\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ converge?

$\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ Please let me know how you did it. Thank you.
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2answers
35 views

Very slow convergence of a particular series?

I've read that $$ \sum_{k=2}^{\infty} \frac{1}{k (\log k)^2} = 2.1097\ldots $$ However when I compute the partial sums it looks like a lot of terms are needed to even get the first decimals right. My ...
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2answers
25 views

Another Divergent Series Question

Suppose the series $$\sum{a_n}$$ diverges where $a_n\ge 0$ and the sequence is monotone non-increasing. If exactly one element is chosen from each interval of size $k$ -- i.e., one element from ...
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1answer
23 views

Divergent Infinite Series Question

If the infinite series $$\sum{a_n}$$ diverges where all terms are positive and $\lim{a_n}=0$, is it always possible to construct a subsequence such that its series converges? That is, does there ...
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28 views

Pointwise convergence and uniform

There is a norm $\Vert .\Vert$ in the space $C([a,b],\mathbb{R})$ such that convergence pointwise implies convergence in norm $\Vert .\Vert$ ? I think not because if there would be the natural ...
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31 views

Strongly convergence in L^2

Let the sequence $(f_n(x,u))$ such as, for all $n,$ $f_n$ is Caratheodory, and $|f_n|\leq g$ where $g \in L^1(\Omega)$ Let $u_n \in H^1_0$ such as it is strongly convergent to $u$ in $L^2$ and a.e ...
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1answer
18 views

Prove this monotone sequence has a bound, thus it converges.

Let $r>0$ and $\frac{r^n}{n!}$ Prove that it converges. I know that it is eventually decreasing, so it is monotone. How do I get a bound for it to show that it converges? Also how would I go ...
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1answer
29 views

Which properties do still hold for the limit of a sequence of functions?

I think it would be very useful to have a list of properties that are preserved for the limit of a sequence of functions, and was wondering if you could help me making the list more complete. Let ...
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4answers
58 views

How to find answer to the sum of series $\sum_{n=1}^{\infty}\frac{n}{2^n} $

I have put his on wolfram and obtained answer as follows: $\sum_{n=1}^{\infty}\frac{n}{2^n} = 2$ And the series is convergent too because $\lim_{n\to\infty} \frac {n}{2^n} = 0$ However I am ...
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1answer
28 views

Prove that this sequence converges

I need to show that $ |r^n|$ converges for $0<|r|<1$. I know this converges to $0$. The problem that I have is that I need to use the observation that $\lim_{x\to inf}|r^{n+1}|=\lim_{n\to ...
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0answers
32 views

Does this condition imply convergence of $\sum_k x_k$?

Let $\{x_n\}_{n \geq 1}$ be a sequence of real numbers (in principle, they could be complex numbers, but I don't think this makes much of a difference for this problem). Suppose the sequence $x_n$ ...
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3answers
37 views

Convergence of $\sum \frac{\sqrt{a_n}}{n^p}$

For $a_n \geq 0$, and $\sum a_n$ convergent, show that $\sum \frac{\sqrt{a_n}}{n^p}$ is also convergent for $p > 1/2$? What bugs me more is why isn't $\sum \sqrt{\frac{a_n}{n}}$ convergent?? ...
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8 views

Differentiability of random processes.

I know the appropriate criterions for mean-square differentiability of random processes. These criterions are connected with covariance function of a process. Are there any criterions for ...
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1answer
26 views

Convergence parameter: Find the value of $p>0$ for which the series converge

For the sum for $k=2$ to infinity: $$\frac{\ln k}{k^p}\ $$ The textbook says the answer is $p>1$.
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1answer
23 views

Convergence almost everywhere

Let consider rational numbers $\{r_n\}_{n=1}^{\infty}$ on [0, 1]. How to prove, that such sum $$\sum_{n=1}^{\infty}\frac{1}{n^2|x-r_n|^{0.5}}$$ converges almost everywhere on [0, 1]. There are my ...
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2answers
26 views

A question about limsup and limif

Could you please help me understand this question: Suppose $a_n$ is bounded sequence and $A<\liminf a_n$, $B>\limsup a_n$. Prove : $A<a_n<B$ for all n>N. It seems to me to simple to be ...
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1answer
37 views

Determine whether the series converge (adding fractions)

$$\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + ... $$ Help convert to summation. Not sure what test to use.
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1answer
28 views

Use comparison or limit comparison test to determine whether the series converge [on hold]

Summation symbol $$\frac{(k^2+1)^{1/3}}{(k^3+2)^{1/2}} \ .$$
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1answer
41 views

Radius of Convergence of a Complex Taylor Series

I've recently been doing some complex analysis questions and come across a few of this type: Find the radius of convergence of the Taylor series at $z=-1$ of the function ...
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1answer
33 views

Prove that the following sequence converges and find its limit

Prove that the following sequence converges and find its limit $$\frac{1}{3+\frac{1}{3}},\frac{1}{3+\frac{1}{3+\frac{1}{3}}},\frac{1}{3+\frac{1}{3+\frac{1}{3+\frac{1}{3}}}},\ldots$$ I started to ...
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1answer
43 views

Identify the radius of convergence from ordinary generating function

Assume we have the ordinary generating function $f(x)$ of a series: $f(x) = \tan x$ Can we identify the radius of convergence for this series?
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50 views

Suppose that $\sum a_i$ converges and that $a_i\geq0$ for all 𝑖.

Suppose that $\sum a_n$ converges and that $a_n\geq0$ for all $n$. For each $n$, let $e_n=\pm1$. Then, prove that $\sum e_na_n$ converges. Can I simply say that ∑|eᵢaᵢ| = ∑aᵢ so that ∑eᵢaᵢ converges ...
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Markov Chain Geometric Convergence

Perhaps due to my non-mathematician background (lack of Measure Theory knowledge), I have some difficulties about the Markov Chain Theory. Given an ergodic (irreducible and aperiodic) Markov Chain ...
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0answers
14 views

Inequality/ Convergence for two operators with functional calculus

Given a sequence of functions $f_n \to f$ in $L^\infty(\mathbb{R}^2)$ and two self-adjoint, unbounded operators $A, B$ is it true that $\|f_n(A,B) - f(A,B)\| \to 0$? With only one operator I can ...
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1answer
28 views

Convergence of $\int_{0}^{\pi} \frac{\sin{(x)}}{(x+n\pi)^{p}} dx$

$$\int_{0}^{\pi} \frac{\sin{(x)}}{(x+n\pi)^{p}} dx$$ where $p>0$ And $n\in\mathbb{N}$. I understand we can compare this to $$\int_{0}^{\pi} \frac{1}{(x+n\pi)^{p}} dx$$ which tells us it converges ...
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1answer
47 views

Finding another series for a given series.

For each sequence $$a_n$$ find a number r such that $$\frac{a_n}{r^n}$$ has a finite non-zero limit. (This is of use, because by the limit comparison test the ...
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Does this recursion/sequence of iterated infinite sums converge?

Let $n,x\in\mathbb{N}$, $\alpha,\beta,\lambda\in\mathbb{R}^+$, where $\alpha,\beta<1$. Does the following sequence converge (and to what)? $s_0=\alpha n+\lambda$ ...
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1answer
20 views

$ \sum_{j=1}^{\infty} \frac{(2^j)+ j}{(3^j) - j} $ converges [duplicate]

Show the series $ \sum_{j=1}^{\infty} \frac{(2^j)+ j}{(3^j) - j} $ converges. I have looked at an answer here, but I do not understand what these results give us. For example, in the first answer: ...
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1answer
14 views

Series convergence or divergence

Determine whether this series converges: $\Sigma e^{-j+sinj}$ I know that this series is $\leq$ than $\Sigma e^{-j+1}$, but I am having trouble getting this in a form appropriate for a convergence ...
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1answer
50 views

Absolute convergence when all the rotated series converge

The question here might be standard in some textbook. Let $a_n, n\ge1$ be a series of real numbers. It is evident that if $\displaystyle \sum_{n\ge 1} |a_n|<+\infty$, then $\displaystyle ...
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1answer
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Find the integral in the complex plane

I'm having some trouble computing these integrals, they're on the practice final, but no solutions given. I'm hoping to get some help here. Calculate the following Integral of $(z \cdot ...
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3answers
116 views

determine if $\sum^{\infty}_{n=1} \frac{n}{e^n}$ converges

determine if $\sum^{\infty}_{n=1} \frac{n}{e^n}$ converges. I used the ratio test and eventually came to that: $\frac{n+1}{en}$ which is approximatley equal to $\frac{1}{e}$ which is less than 1. ...
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Need help to understand calculating fourier transform of 1

this issue is related to a physics problem, but since it is mathematical I will post it here. When calculating the following Fourier transform $$ -i\int_\infty^\infty dt~ e^{i\omega ...
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35 views

Bound on $f_n'$ implies uniform convergence of $f_n$?

Let $f_n$ be a sequence of functions that converge pointwise to a function $f$. Suppose I know that $|f_n'(x)| \leq C(x)$ where the constant doesn't depend on $n$. How do I conclude that $f_n \to f$ ...
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1answer
57 views

Identify the radius of convergence [on hold]

For the ordinary generating function $f(z) = \frac{z^3 +1 }{z^3 -1}$, how can we identify its radius of convergence? And is this function meromorphic?
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2answers
51 views

Does differentiability imply convergence

Can we say that if the limit of a sequence of functions is differentiable then the sequence is convergent? I mean, I know that $\frac{\partial f(x,t)}{\partial x}$ exists. If I specify a sequence ...
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3answers
29 views

Convergence of an Alternating Series

For this question, I`m trying to determine the values of $p$ in which the series converges. The series is: $\sum_{n=1}^\infty \frac{(-1)^n}{n^p}$. I already know (by looking at the answer to the ...
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Series convergent but not absolutely? $\sum_{n=1}^{\infty} \frac{\cos(n^p \pi)}{n^p}$

For which real numbers $p>0$ does the series $$\sum_{n=1}^{\infty} \frac{\cos(n^p \pi)}{n^p}$$ converge? Obviously it converges absolutely for $p>1$ but what about $0<p<1$? I have the ...
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1answer
23 views

Is this set measurable? (Set of points where a sequence converges)

Let $M$ be a manifold. Suppose that $u_n:M \to \mathbb{R}$ are measurable and we have $u_n(s) \to u$ a.e. in $M$. Does it follow that the set $A=\{s \in M : u_n(s) \to u(s)\}$ and $A^c$ are ...
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An example of a sequence of continuous real functions pointwise convergent, but nowhere locally uniformly convergent? [duplicate]

I've been trying to come up with an example of a sequence of continuous real function which would converge pointwise everywhere, but nowhere converge locally uniformly, but I can't really think of ...
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48 views

Convergence in probability implies convergence in mean under one additional condition

Prove that if random variables $X_n$ are dominated by an integrable random variable then $E[X_n] \to E[X]$ follows if $X_n$ converges to $X$ in probability. Hint: Use the following theorem : A ...
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2answers
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How to prove convergence of $a_n$ if $(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$?

Could you give me some hint how to conclude convergence of $a_n$ from this feature : $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ From $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ we may conclude that ...
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3answers
75 views

Simple series convergence/divergence

I have the following problem: $$\sum_{k=1}^{\infty}\frac{2^{k}k!}{k^{k}}$$ I only need to find whether the series converges or diverges. My initial thinking was to use the ratio test. I hit a stump ...
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2answers
29 views

Convergence of an improper integral(with parameters)

I'm trying to find solution to this problem: For what pairs (a; b) of positive real numbers does the improper integral $$ ...
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1answer
56 views

Prove that $\varphi^n(t) \rightarrow 0$ when $n \rightarrow \infty$

I have to prove a lemma: If $\varphi: \mathbb{R}_+ \rightarrow \mathbb{R}_+$ is monotone increasing and $\varphi(t) < t, \ \forall t \in \mathbb{R}_+$, then $\varphi^n(t) \rightarrow 0$, $(n ...
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1answer
35 views

Convergence of series of continuous bounded functions

Let $\mathcal{C}$ be the set of continuous bounded mappings from $\mathbb{R}^n$ to $\mathbb{R}^m$. Let $F: \mathcal{C} \rightarrow \mathcal{C}$ be a continuous (with respect to the $\sup$ norm) ...
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2answers
39 views

Testing A Series For Convergence

Determine whether the series $\sum_{n=0}^{\infty} \frac{3n^2 + 2n + 1}{n^3 + 1}$ with n from 0 to infinity converges or diverges. So far I thought about dividing the numerator by the denominator, ...
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4answers
128 views

Why do some series converge and others diverge? [closed]

Why do some series converge and others diverge? What causes the divergence or convergence of a series and why does that cause such a behavior? For example, why does the harmonic series diverge, but ...