Convergence of sequences and different modes of convergence.

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26 views

Which functions are irrelevant?

Call a bijection $f : \mathbb{N} \rightarrow \mathbb{N}$ irrelevant over $\mathbb{R}$ iff for all sequences $a : \mathbb{N} \rightarrow \mathbb{R}$, if $$\sum_{i=0}^\infty a_n$$ exists, call its value ...
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0answers
9 views

Limit laws for power of convergent sequence

Let $ \{ a_n \}_{n=m}^{\infty}$ be convergent sequence of real numbers such that $\lim _{n \rightarrow \infty}a_{n}=x$. Can we say that $\lim _{n \rightarrow \infty}a_{n}^q=(\lim _{n \rightarrow ...
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1answer
21 views

For what values of $k$ to both of the following series converge?

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
0
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0answers
9 views

Newton Method Variant with convergence of order 3

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be twice continuously differentiable for all $x$ in the neighborhood $U(\xi)=\{x\in\mathbb{R}:|x-\xi|<r\}$ of a simple zero $\xi$ of $f$ such that ...
1
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0answers
19 views

For which $s$ is defined $\frac{1}{\zeta(s)}\sum_{n=1}^\infty\frac{n}{ \left( \zeta(s) \right) ^n}$, where $\zeta(s)$ is the Riemann Zeta function?

If there are no mistakes in my words, if one of the factors $\sum_{n=1}^\infty a_n$ of a convolution product or Cauchy product converges, and the other $\sum_{n=1}^\infty b_n$ corverges absolutely ...
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1answer
18 views

Sequence converging to a closed set

How can I prove that A sequence generated by a particular algorithm converges to a set? i.e irrespective of the starting point the sequence converges to any of the points in a set, or oscillates ...
0
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1answer
25 views

Determine whether the following sequences (fn) converge uniformly, pointwise, or neither:

Determine whether the following sequences $(f_n) \in F(E, \mathbb{R})$ - where E is a set - converge uniformly, pointwise, or neither: a) $f_n(x) = \frac{n^2x} { 1 + n^2x^2}$ on set $E = \mathbb{R}$ ...
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5answers
46 views

How to prove that $\lim\limits_{n\to\infty}\sqrt[n]{p}=1$? [on hold]

Could you tell me how to show if $p>0$ then$\lim\limits_{n\to\infty}\sqrt[n]{p}=1$? (+clues) 1.put $\sqrt[n]{p}=1+h_{n}$ 2.Bernoulli's inequality If you don't mind, use the clues to prove it.
0
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2answers
567 views

Confused about series and testing for convergence/divergence?

I'm finding it quite difficult to understand the idea of series and limits to test for divergence or convergence. Perhaps more so in finding such a limit. I have the series $$\sum_{n=1}^\infty ...
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0answers
6 views
-1
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0answers
24 views

A non unconditionally convergent series? [on hold]

A series $\sum_{n\ge 1}a_n$ in a Banach Space $X$ is said to be unconditionally convergent if it converges for any rearrangement of its terms. Denote by $(e_n)$ the canonical basis of $\ell_2$, ...
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2answers
45 views

Investigate convergence of $\sum_{n=1}^\infty \frac{\ln(n)}{n}$

Investigate convergence of $\sum_{n=1}^\infty \frac{\ln(n)}{n}$ I applied nth term test and was inconclusive. I tried ratio test but I don't know how to evaluate the limit. I think it is 1 therefore ...
0
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1answer
33 views

Radius of Convergence of $\sum_{n = 0}^{\infty} \frac{(-1)^nn!x^n}{n^n}$

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
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3answers
72 views

Is convergence in probability sometimes equivalent to almost sure convergence?

I was reading on sufficient and necessary conditions for the strong law of large numbers on this encyclopedia of math page, and I came across the following curious passage: The existence of such ...
0
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1answer
18 views

Prove point-wise convergence for the sequence $\{f_k\}$ of functions

Consider the sequence $\{ f_k\}$ of functions in $C[0,1]$ defined by $$ f_k(x) = \begin{cases} 0, & 0 \leq x \leq \frac{1}{k} \\ 2(k^{3/2} x - k^{1/2}), & \frac{1}{k} \leq x \leq ...
1
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1answer
30 views

Find convergence domain of the integral

Find convergence domain of $$\int_0^\infty \! \frac{\cos^2{x}}{x^p} \, \mathrm{d}x$$ I've tried to use $\frac{\cos^2{x}}{x^p} < \frac{1}{x^p}$, but $\int_0^\infty \! \frac{1}{x^p} \, \mathrm{d}x$ ...
2
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0answers
7 views

When does the cohomological Atiyah–Hirzebruch–Leray–Serre spectral sequence converge?

Given a Serre fibration $F \to E \to B$ of spaces homotopy equivalent to CW complexes, with $B$ simply-connected, and a generalized homology theory $h_*$ with respect to which the fibration is ...
0
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1answer
36 views

Investigate convergence of $\sum_{n=2}^\infty \frac{1}{n(n-1)}$

Investigate convergence of: $\sum_{n=2}^\infty \frac{1}{n(n-1)}$ Now I know by $p$-series test this summation converges however, is there a way to prove that this series converges by some ...
4
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0answers
50 views

Can the boy escape the teacher for a regular $n$-gon?

This is related to Prove that the boy cannot escape the teacher Suppose there is a boy in the center of a regular $n$-gon. The teacher is on the edge of the $n$-gon (but cannot leave the edge) and ...
3
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2answers
45 views

Determine wether $\sum_{n=1}^\infty \frac{3n-1}{(n+1)^3}$ converges or diverges.

Determine wether the following function converges or diverges by comparison test: $\sum_{n=1}^\infty \frac{3n-1}{(n+1)^3}$ Upon inspection I can clearly see that the series converges. However I am ...
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1answer
28 views

Proof of a theorem regarding subsequences and convergence

I am attempting to understand the following theorem and it's proof as outlined in my textbook for Real Analysis. Theorem: Let $(s_n)$ be a sequence. If t is in $\mathbb{R}$, then there is a ...
6
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1answer
36 views

$\mu(X) \lt \infty$. Then $f_k \to f$ in measure iff for any subsequence $k_l$, there is a subsequence $k_{l_n}$ such that $f_{k_{l_n}}\to f$ a.e.

Let $\mu(X) \lt \infty$. Then $f_k \to f$ in measure iff for any subsequence $k_l$, there is a subsequence $k_{l_n}$ such that $f_{k_{l_n}}\to f$ a.e. I can show the only if part by using the theorem ...
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2answers
71 views

Solving equations like $xe^x = c$ via functional iteration

Yesterday I randomly thought of solving $xe^x = c$ via functional iteration (FI) after manipulating the equation into a form "$x = \cdots$" that gives the 'fastest' convergence rate regardless of the ...
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24 views

series convergence help using tests [on hold]

Use any theorems or properties of series. This was a question on my homework and I received 0 points. I need help with the entire question. I originally tried to compare part (a) to the ...
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1answer
25 views

Let $f_k \to f$ in $L^{\infty}$, then $f_k \to f$ in measure.

Let $f_k \to f$ in $L^{\infty}$, then $f_k \to f$ in measure. I looked at the proof of this statement and it says that it follows from the fact that if $f_k \to f$ in $L^{\infty}$, then $f_k \to f$ ...
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1answer
13 views

Give necessary and sufficient conditions so the sum of random variables converges almost surely

$\{X_k\}_{k}$ are independent random variables on the probability space $(\Omega, \mathcal F, P)$ and $X_k$ has gamma density $f_k(x)$ where $f_k(x)=\dfrac{x^{a_x-1}e^{-x}}{\Gamma(a_k)}$ where $x,a_k ...
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1answer
28 views

Find the radius of convergence of $\sum_{n=1}^\infty n!(2x-1)^n$

Find the radius of convergence of $\sum_{n=1}^\infty n!(2x-1)^n$ Now, by D'Alemberts Ratio test that implies (for convergence): $\lim_{n\to\infty} \lvert \frac{(n+1)!(2x-1)(2x-1)^n}{n!(2x-1)^n} ...
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1answer
21 views

Find convergence domain of integral

I need to find convergence domain of $$\int_1^2 \! \frac{\ln(x-1)}{(4-x^2)^p} \, \mathrm{d}x$$ I've tried to use estimates like $\frac{\ln(x-1)}{(4-x^2)^p} < \frac{1}{(4-x^2)^p}$ and change of ...
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0answers
24 views

Show that $C^1([a,b])$ is a complete space w.r.t. some special metric [duplicate]

Let $C^1([a,b])$ denote the space of all continously differentiable functions on $[a,b], a,b\in\mathbb{R}$. On this space, define the following metric: $$ d(f,g)=d_{\infty}(f,g)+d_{\infty}(f',g'), $$ ...
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3answers
35 views

Prove that functional series $\sum_{n=1}|sinx|^{\sqrt{n}}$ is convergent when $x \in (-1,1)$

Prove that functional series $\sum_{n=1}|sinx|^{\sqrt{n}}$ is convergent when $x \in (-1,1)$. I do not know if that's so easy that I'm simply missing something, but I can't find any criterion which ...
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0answers
25 views

find $\text{limsup} \dfrac{X_n}{\ln{n}}$? how can i apply Borel-Cantelli here?

let $(X_n)_{n\geq1}$ be a sequence of independent random variables. Suppose that the density function of $X_n$ is: $$ f(x)=\dfrac{1}{2}.e^{-|x|} \quad x \in \mathbb{R} \quad \forall n \quad ...
2
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3answers
45 views

Rigorous Definition of One-Sided Limits

In a typical first-year Calculus course professors typically tend to put a lot of emphasis on making visual connections when working with "one-sided" limits or derivatives. This is something I find ...
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3answers
32 views

Proof about infinite limsups and subsequences

I am working on a final exam study guide, and came across this question: Suppose limsup$(a_n)$ = $\infty$. Prove: There must exist a sub-sequence ${a_n}_k$ such that ${a_n}_k \to \infty$. My initial ...
2
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1answer
34 views

Integrating variation of error function: $\int_1^2e^{-nx^2} dx$

Show that $$\lim_{n\to\infty} \int_1^2e^{-nx^2} dx = 0.$$ After much googling, I learned that I am working with a variation of the error function! Yay. I've never heard of it in my life and I ...
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0answers
7 views

Understanding the uniform convergence of $\sum_{m\geq 1}\frac{z(z-1)}{2m^2}$

Define $a_m(z)=\frac{z(z-1)}{m^2}$, for all $z\in \mathbb{C}\setminus \{-1,-2,\dots\}$. The sum $\sum_{m\geq 1}a_m(z)$ clearly converges absolutely. Is there a fair approach to prove that it ...
0
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1answer
37 views

Shouldn't all alternating series diverge by the diverge test?

An Alternating Series, as defined in my textbook, is of the form $\sum (-1)^n b_n$. If we look at the nth term, the series doesn't appear to converge. If n is odd, the nth term is negative; if it's ...
2
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0answers
17 views

Prove $\sum_{k=1}^\infty k^{-p}f(kx)$ converges absolutely almost everywhere, where $p>0, f \in \mathcal{L}^1(\mathbb{R})$.

What I've done: $$ \int_\mathbb{R} \sum_{k=1}^\infty k^{-p}|f(kx)| = \sum_{k=1}^\infty \int_\mathbb{R} k^{-p}|f(kx)|dx = \sum_{k=1}^\infty k^{-p}\int_\mathbb{R} k^{-1}|f(y)|dy = ...
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1answer
25 views

Relation between $\lim_{n \to \infty}\int_{I}f_n(x)\:dx$ and $\int_{I}\lim_{n \to \infty}f(x)\:dx$ [on hold]

Relation between convergence and integration of sequence of a function. Let $f_n$ be a sequence of integrable functions defined on an closed interval with $$f_n(x) \to 0$$ on this interval ...
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0answers
22 views

show that $U_n$ converges to $0$ in $L^1$ and almost surely.

let $(X_n)_{n\geq1}$ be a sequence of independent random variables. Suppose that the density function of $X_n$ is: $$ f(x)=\dfrac{1}{2}.e^{|x|} \quad x \in \mathbb{R} \quad \forall n \quad ...
2
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0answers
18 views

Almost sure convergence and limes superior

I'm trying to prove the following exercises and I don't know if my attempts are correct. A sequence of real random variables $(X_n)$ almost surely converges to $X$ if and only if for every $\epsilon ...
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1answer
56 views

$a_n = b_n -b_{n-1}$ Prove that $\sum_{n=1}^{\infty} a_n$ converges iff $\lim_{n \to \infty} b_n$ exists

Let $\{b_n\}$ be a sequence Let $a_n = b_n - b_{n-1}$. Prove that $\sum\limits_{n=1}^{\infty} a_n$ converges iff $\lim_{n \to \infty} b_n$ exists. I am extremely stuck on this homework problem and ...
2
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2answers
29 views

Showing a recursively defined sequence is convergent

For $a_1=1$ and $a_{n+1} = 1 + \frac{a_n}{3+n}$, I want to show that the sequence $a_n$ converges. I will use the Monotone Convergence Theorem. Of course, the sequence is bounded below by $1$. Now I ...
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2answers
43 views

Theorem 3.19 in Baby Rudin: The upper and lower limits of a majorised sequence cannot exceed those of the majorising one

Here is Theorem 3.19 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. Let $\{s_n \}$ and $\{t_n \}$ be sequences of real numbers. If $s_n \leq t_n$ for $n \geq N$, ...
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4answers
54 views

Why does the sequence $a_n = (n^2)(1 - \cos(\frac{4.4}{n}))$ converge to 9.68?

Find the limit of the sequence whose terms are given by $a_n = (n^2)(1 - \cos(\frac{4.4}{n}))$. The given answer for this problem is $9.68$. What rules about sequences, and steps, should be taken ...
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0answers
27 views

Is there a way to calculate RMS value continuously?

Using that the RMS by definition is: $\sqrt {\int_0^T\frac 1T*f(t)^2dt} $ which can be calculated by using Riemann sums in the following way: $\sqrt {\frac 1N\sum_0^Nf[i]^2} $ I've tried that in ...
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1answer
52 views

Examine convergence and (almost) uniform convergence of $\sum_{n=1}^{+ \infty} \frac{n^2 x^2}{e^{n^2 |x|}}$ for $x \in \mathbb{R}$.

How to examine convergence, almost uniform convergence and uniform convergence of series $\sum_{n=1}^{+ \infty} \frac{n^2 x^2}{e^{n^2 |x|}}$ for $x \in \mathbb{R}$?
2
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1answer
63 views

Prove that $\frac{n^n}{(n + 1)^n}$ converges to $\frac{1}{e}$

Using the formal definition of sequence limits, I would like to prove that for: $$ a_n := \frac{n^n}{(n + 1)^n} $$ it is: $$ \lim_{n\to\infty} a_n = \frac{1}{e}. $$ Thus, it remains to show ...
0
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0answers
70 views

Is $n^{1/10}$ a cauchy sequence?

So, I stuck at a step to prove that $n^{1/10}$ is not a cauchy sequence. So, $$|y_n - y_m| = |n^{1/10} - m^{1/10}|$$ So, now how is the next step to show, that it is not a cauchy sequence?
2
votes
0answers
18 views

Theorem 3.17 in Baby Rudin: Infinite Limits and Upper and Lower Limits of Real Sequences

Here're Definitions 3.15 and 3.16 and Theorem 3.17 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. Definition 3.15: Let $\{s_n \}$ be a sequence of real numbers ...
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1answer
47 views

Prove $\sum_{n = 0}^{\infty} a_n X^n$ converges at every point in $[-1,1)$ if $a_n$ is non-increasing and converges to $0$

Prove that if $a_n$ is non-increasing, converges to $0$, and radius of convergence of $\sum_{n = 0}^{\infty} a_n X^n$ is equal to $1$ then this series converges at every point in $[-1,1)$ I am trying ...