Convergence of sequences and different modes of convergence.

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Find the Interval of convergence, help?

Find the convergence interval of the series: well it goes to infinity and $n=0$ of $n^3(5x+10)^n$ :) EDIT: I solved it, I had to use the ratio test and endt up with 5x+10<1 and then just solved ...
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45 views

Bounded function

${f_n}$ is a sequence of continuous functions on $\Bbb R$, and $f_n \rightarrow f$ uniformly on every finite interval $[a,b]$. If each $f_n$ is bounded, is it true that $f$ must be bounded?
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3answers
56 views

Show the following series converges or diverges using the comparison test.

The series is: $\sum_{n=1}^\infty {(\frac{n}{n+2})}^{n^2}$. I'm relatively certain it converges, but I'm not sure how to prove this. It was suggested that I try and use the comparison test, but I'm ...
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442 views

How prove this integral $\int_{0}^{\infty}f^{\alpha}(x)dx,\alpha>1$ is convergent

Question: let the function $f(x)\ge 0$,and such $$f'(x)\le\dfrac{1}{2},\forall x\ge 0$$ and this integral $\displaystyle\int_{0}^{\infty}f(x)dx$ is convergent. show that: ...
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1answer
77 views

Strong convergence of bounded sequences in Bochner spaces

Let $S=(0,T)$ for a $T>0$ and let $B_0,\ B_1,\ B_2$ be Banach spaces, such that $B_0$ is compactly embedded in $B_1$, which is in turn continuously embedded in $B_2$. Suppose we have a sequence ...
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73 views

How do I calculate the Radius of convergence of this sum [closed]

What is the radius of convergence? $$\sum_{n=0}^{\infty} n^3 (5x+10)^n$$
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1answer
66 views

Convergence of product over all primes

How can we find the values of $x$ for which $$\prod_{p \text{ prime}}{1-\frac{x^2}{p^2}}$$ converges? I know that this product $$\prod_{p \text{ prime}}{1+\frac{x^2}{p^2}}$$ converges if and only if ...
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6answers
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Prove that $x^n/n!$ converges to $0$ for all $x$ [duplicate]

Prove that $a_n=x^n/n! \to 0$ for all $x$ Here is what I tried, but it seems to lead to nowhere. Choose $\epsilon > 0$. We need to show that there exists $N\in \mathbb{N}$ such that for all ...
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1answer
92 views

Variant of dominated convergence theorem

There are several variants of dominated convergence theorem. The standard one requires $f_n \to f$ a.e. and $|f_n|\leq g$ a.e. where $g$ is integrable. It can be weakened to only convergent in ...
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1answer
186 views

About the epsilon definition of a convergent sequence. Is this definition equivalent?

I read that it is appreciated to include the context and motivation of a question. I may have overdone this a little bit in this question. To summarize, my question is: Are the two blockquoted ...
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0answers
33 views

Convergence proof of non convex formulation

Assume that we have a non-convex optimization $\min_{A,B} f(A,B)+\lambda g(A,B)$. Specifically, $f(A,B)+\lambda g(A,B)$ is not joint convex, but it is convex with regard to one variable when fixing ...
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1answer
332 views

Convergent sequence in Lp has a subsequence bounded by another Lp function

For $E$ a measurable set and $1\leq p<\infty $, assume $f_n\to f$ in $L^p(E)$. Show that there is a subsequence $\{f_{n_k}\}$ and a function $g\in L^p(E)$ such that $|f_{n_k}|\leq g$ almost ...
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2answers
63 views

Is $Z_n→Z$ in distribution and if $z_n→0$ then $z_nZ_n→0$ in distribution

I'm trying to prove the statement made by Did in the comments: Is $Z_n→Z$ in distribution and if $z_n→0$ then $z_nZ_n→0$ in distribution. So we need to prove that $$\forall t>0: ...
2
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1answer
162 views

sequence uniformly convergent on the boundary of a bounded set in $\mathbb{C}$

Let $(f_n)$ be a sequence of functions which are analytic on a bounded region $A\subset\mathbb{C}$ and continuous on the closure $Cl(A)$. Suppose that the sequence is uniformly convergent on the ...
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2answers
55 views

A function is a.e. equal to a polynomial.

Let $f\in{L^p}$. For $t>0$, let $P_{t,n}(x)$ be a collection of polynomials of degree less then or equal to $n$ in the variable $x$ and the family is given by $t$ such that ...
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Does this sequence of operators converge in norm or strongly?

Let $H$ be a Hilbert space and $\mathcal{L}(H)$ the set of all bounded linear operators $L:H\to H$, equiped with the usual norm $\|\cdot\|_{\mathcal{L}}$. Let $T:D(T)\subset H\to H$ be a ...
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1answer
162 views

Improper integral convergence from minus to positive infinity

Quote from Essential Calculus: Early Transcendentals, by James Stewart: If $f$ is continuous, then $$\int_{-\infty}^\infty f(x)dx=\lim_{t \to \infty}\int_{-t}^tf(x)dx$$ I thought this would be ...
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1answer
71 views

weak-*-convergence of measures ==> convergence of the total mass?

Let $X = [0,1]$. Let $\mu_n$ be a sequence of regular signed Borel measures on $X$, which converges to a measure $\mu$ on $X$ in weak-star, i.e. for any $f\in C_0(X)$, we have $\int_X f \mu_n(dx) \to ...
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4answers
104 views

Does the following series converge or diverge?

(2n-1)/(n!) I used the ratio test here and got the lim as n -> infinity to be 0. Therefore, I assumed that the series converges. However, my textbook says that it ...
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66 views

Determining whether the series $\sum_{n=1}^\infty \frac n{n+1}$ converges

I have this series: $$\sum_{n=1}^\infty \frac n{n+1}$$ I have to determine if this series converges. If I do the ratio test, I get: $$L = \lim_{n\to\infty} ...
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1answer
84 views

Showing if $f_n \to f$ uniformly and each $f_n$ has at most $10$ discontinuities, then so does $f$

Suppose that $f_n:[a,b] \to \Bbb R$ and $f_n$ uniformly converges to $f$ as $n$ goes to infinity. How to prove that if each $f_n$ has at most ten discontinuities (the discontinuities for each $f_n$ ...
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Convergent integral of divergent function

On one of my calculus lectures i was told that exist convergent inmproper integrals (in infinity) of divergent function. I was searching for an example in the internet, but I didn't find any. Has any ...
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1answer
74 views

Find the radius of convergence of the power series

$\displaystyle\sum_{n=0}^{\infty}a_nz^n$, where $a_{2k+1} = 2^k$ and $a_{2k} = (1 + (1/k))^2$ for $k = 0, 1, 2, \dotsc$ I started off by doing the ratio test, but I know that the ration test is for ...
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0answers
54 views

Showing how this infinite sum diverges [duplicate]

$\displaystyle{\sum_{n=1}^{\infty} \left[\left( 1 + {1 \over n}\right)^{n} - {\rm e}\right]}$ I tried both root and ratio tests (for the root test, the expression became way too complex to handle) ...
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1answer
63 views

How to make use of the hint for proving $\text{CLT} \implies \text{WLLN}$?

I've seen an exercise where one is asked to prove that the central limit theorem implies the weak law of large numbers. The author gave the following hint: "First prove that convergence in ...
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64 views

Estimation for sum over binomial coefficients

I am trying to show that a certain procedure for resource allocation is approximately efficient. For this I need to show that $$ \lim_{n\rightarrow \infty} \left(\frac{1}{e}\right)^n\sum_{c=2}^n ...
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1answer
81 views

Three questions about ucp convergence

We say that a sequence of processes $X^n$ converges to a process $X$ uniformly on compacts in probability if for all $\epsilon >0, t>0$ $$P[\sup_{s\le t}|X^n_s-X_s|>\epsilon]\to 0 $$ for ...
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terms $(x^n/n!)$ approach 0 faster than $(1/x)^n$

Can anyone help me prove that the terms $\frac{x^n}{n!}$ approach $0$ faster than $(\frac{1}{x})^n$. Thank you in advance for the help.
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Show that If $\sum_{n=1}^\infty b_{n}$ is abosolutley convergent, then $|\sum_{n=1}^\infty b_{n}| \leq | \sum_{n=1}^\infty |b_{n}|$

this problem was given as a practice problem for first year calculus class. Here it is: show that if the series $\sum_{n=1}^\infty b_{n}$ is abosolutley convergent, then $|\sum_{n=1}^\infty b_{n}| ...
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2answers
45 views

Prove that if $\int_a^\infty g(x) dx$ is convergent then $\int_a^\infty f(x) dx$ is convergent.

where $f$ and $g$ are positive continuous functions on $[a, \infty)$, and $$\lim_{x\to\infty} \frac{f(x)}{g(x)} = 0$$ I tried to prove this as follows: But something tells me I can't do the ...
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62 views

Determine interval of convergence and the sum

Determine interval of convergence and the sum: $$ \sum_{n=0}^\infty \frac{x^n}{n+3} $$ any tips?
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2answers
369 views

If $\,x>1$, then $\displaystyle\lim_{n\rightarrow\infty}\frac{\left\lfloor x^{n+1} \right\rfloor}{\left\lfloor x^n \right\rfloor}=x$.

How can I prove that $$ \lim_{n\rightarrow\infty}\frac{\left\lfloor x^{n+1} \right\rfloor}{\left\lfloor x^n \right\rfloor}=x, $$ whenever $x>1$. Here $\left\lfloor \cdot\right\rfloor$ denotes the ...
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36 views

Convergent sequence proof

When asked to prove that $\sum\limits_{i=1}^{\infty}{{F_i} \over {10^{i+1}}}$ , where $F_i$ is fibonacci sequence, converges to a rational number, am I right to assume that it is satisfactory to prove ...
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1answer
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Determining a series convergence using root test

I just have a few quick questions on using the root test to determine the convergence a series. If I get a series $\sum_{n=5}^\infty A_n$. What would the $n=5$ do? From my knowledge when I apply root ...
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147 views

Prove that if $\sum_{n=1}^{\infty} |a_n|$ converges and $(b_n)^{\infty}_{n=1}$ is a bounded sequence, then $\sum_{n=1}^{\infty} |a_nb_n|$ converges

this was given as an exercise: Prove that if $\sum_{n=1}^{\infty} |a_n|$ converges and $(b_n)^{\infty}_{n=1}$ is a bounded sequence, then $\sum_{n=1}^{\infty} |a_nb_n|$ converges This is what i was ...
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Prove that if $\sum_{n=1}^\infty a_{n}$ is absolutely convergent, then $|\sum_{n=1}^\infty a_{n}| \leq | \sum_{n=1}^\infty |a_{n}|$

Hey everyone this was given as a practice problem for my first year calculus class and it really giving me a headache, any help is appreciated! Prove that if $\sum_{n=1}^\infty a_{n}$ is absoultley ...
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34 views

Evaluate sequences of integrals with function bounded.

Evaluate $\lim\limits_{n\to\infty}n\int\limits_0^1 f(x)e^{-nx}dx$ where $\,f$ is bounded in $\mathbb{R}^+\cup\{0\}$. My problem is that I think there's information missing about $f$, e.g. some ...
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1answer
27 views

Why is this set an event?

As a part of a setup for another problem, my text remarks that it can be used without a proof that if $X_1, X_2, \ldots$ are random variables then $$C:=\{ \omega\in\Omega \ | \ \sum X_n(\omega) \ ...
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Multiplication of infinite series

Why multiplication of finite sums $(\sum_{i=0}^n a_i)(\sum_{i=0}^n b_i)=\sum_{i=0}^n (\sum_{j=0}^ia_jb_{i-j})$ (EDIT: This assumption was shown to be false) does not work in infinite case? I have ...
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How to prove that $\sum_{n=1}^{\infty}\frac{1!+2!+\cdots+n!}{(2n)!}$ converges?

Show that this series converges: $$\sum_{n=1}^{\infty}\dfrac{1!+2!+\cdots+n!}{(2n)!}$$ My solution: this series converges since $$1!+2!+\cdots+n!\le n!+n!+\cdots+n!=n\cdot n!$$ and since ...
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1answer
73 views

How can we show that the following sequence converges?

$(a_n)$ is a bounded sequence with the following condition $a_{n+1}\geq a_n-\frac{1}{2^n}$ The sequence converges, but how do we show it?
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1answer
44 views

How does one show that the series converges almost surely?

Let $X_1, X_2, \ldots:\Omega\to \mathbb R$ be random variables. Define $C:=\{ \omega \ | \ \sum X_n(\omega) \text{ converges} \}$. There is such $q\in(0,1)$ that for all $n\in \mathbb N: P\{ |X_n| ...
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73 views

Prove that the series converges

Let $(a_n)$ be a bounded positive and monotone increasing sequence. I need to show that $\sum (1-\frac{a_n}{a_{n+1}})$ converges. My approach was as follows: Let $B=sup (a_n)$ Then since the ...
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77 views

Taylor series expansion and the radius of convergence

Hello I have some problems concerning Taylor series. Given the function $$f(x)=e^{\sin{x}} $$ I concluded that the Taylor series expansion would be $$f(x) = ...
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1answer
47 views

Topology of a convergence space

I am actually having an introduction to filters. Today I was trying to prove that the collection of open sets of a convergence space satisfy the axioms of a topology: O $\subset$ X is open iff $lim ...
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1answer
73 views

Convergence of Random Variables in mean

If $$E[|X_n-X|^r]\rightarrow0$$ prove that $$E|X_n^r|\rightarrow E|X^r| $$ for every $r\ge 1$ This is the very notation used. I believe it should be: $$E[|X_n|^r]\rightarrow E[|X|]^r $$ Attempt I ...
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1answer
137 views

Why is lower semicontinuity important for epi-convergence?

Why is the lower semicontinuity property important for epi-convergence (and, on the contrary, upper semicontinuity is not desirable)? A simple example would also help.
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914 views

Find the exact value of the infinite sum $\sum_{n=1}^\infty \{\mathrm{e}-(1+\frac1n)^{n}\}$

How can we find the exact value of the infinite sum $$ \displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\Big(1+\frac1n\Big)^n\right\}? $$ This problem appears in: T. Andreescu, T. Radulescu & V. ...
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1answer
90 views

A different type of Convergence of Fourier Series

I have just started studying fourier series. All the convergences I have seen considered the partial sums to be $\sum\limits_{i=-n}^n a_n Sin(n\theta)$. But in all practical systems the harmonics ...
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2answers
72 views

Does the Leibniz Test say that if the sequence doesn't go to 0 that the series diverges?

My book explains the Leibniz test by saying: "Assume a sub n is a positive sequence that converges to 0..." And goes on to say that that means the alternating series converges. What if the sequence ...