Convergence of sequences and different modes of convergence.

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2answers
21 views

Decide if the following series is converge [closed]

Im trying to prove if the following series is converge - I tried the ratio test but couldnt calculate the limit - $$\sum_{i=1}^\infty \frac{(2n)^{n+2}}{(n+1)!}$$ Thanks for helping!
0
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2answers
66 views

Bessel's function in spectral geometry [closed]

I have to compute $Z(t)=\sum_{n=1}^\infty e^{-\lambda_nt}$, $t \in \mathbb{R}_{>0}$, with $\lambda_n=\pi^2(\frac{m^2}{a^2}+\frac{n^2}{b^2})$. So $\sum_{m,n=1}^\infty ...
5
votes
1answer
40 views

Norms inequality in a sequence space

Let $1 \leq p<q \leq \infty$ (p an q are not related) Let $\Phi$ be the vector space of all sequences with at most finitely many nonzero elements, meaning $\Phi=\{\{x_n\}_{n=1}^\infty|$ there is ...
1
vote
1answer
23 views

Is there a rearrangement theorem for conditionally convergent improper integrals?

The famous Riemann rearrangement theorem states that for a conditionally convergent real number series, we can rearrange the order of summation to make it converge to any prescribed number in the ...
0
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1answer
34 views

Uniformly bounded sequence of analytic functions in the unit disk

Suppose $\{f_n\}$ is sequence of analytic functions that is uniformly bounded in the open unit disk and for every positive integer $k$, $f_n(\frac{1}{1+k})\to 0$ pointwise. Then, $\{f_n\}$ converges ...
0
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1answer
14 views

Convergence in mean square and almost surely

Given the sample space [0,1] and the uniform probability measure P(.), random variables $(X_n)_{n\geq1}$ are defined by How do I $X_n$ converges almost surely as n tends to infinity and also in ...
1
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2answers
37 views

Convergence of the sequence $2n\times\sin(\frac{1}{n-1})$?

I have the sequence $$\{2n*\sin(\frac{1}{n-1})\}$$ and I'm supposed to see if this sequence converges. By transforming this sequence into a function of $f(x)$ and applying L'Hopital's rule to the ...
0
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0answers
18 views

How to formally justify matrix manipulation in countable-state Markov chain

I have a Markov chain with transition probabilities $t_{i,i+1} = \binom{k+i}{k}^{-1}$ and $t_{i,0} = 1-t_{i,i+1}$, i.e. we have an absorbing chain with absorption probability approaching one as $i ...
0
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0answers
15 views

Understanding the Strong Law of Large Numbers

Strong law of large numbers (SLLN) says if $X_1, X_2, \dots$ are iid random variables with expectation $\mu$, then $\bar{X}_n \to \mu$ almost surely, or $$P(\lim_{n\to \infty} \bar{X}_n = \mu)=1.$$ ...
0
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1answer
16 views

Convergence of $\int_0^1 x^p ln^q \left(\frac{\ 1}{x}\right)$

So far, I determined that the integral converges for every $q>p+1$. I noticed that for example for the values $p=5, q=3$ the integral still converges. There are some values for which the integral ...
1
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2answers
36 views

Does $\sum_{n=1}^\infty \left(\frac{1}{n} - 1\right)^n$ and $\sum_{n=1}^\infty \left(\frac{1}{n} - 1\right)^{n^2}$ converge?

My task is this: Determin whether $\sum_{n=1}^\infty \left(\frac{1}{n} - 1\right)^n$ and $\sum_{n=1}^\infty \left(\frac{1}{n} - 1\right)^{n^2}$ converge or diverge. My thoughts: For large $n$ one ...
3
votes
1answer
80 views

How do i show $\sum_{n=0}^\infty a_n$ converges iff $\sum_{m=0}^\infty 2^ma_{2^m}$ converges?

My task is this: (i) Let $\{a_n\}$ be a positive decreasing sequence. Show that:$$2^ma_{2^m}\geq a_{2^m} + a_{2^m+1}+a_{2^m+2}+\ldots+a_{2^{m+1}+1}\geq2^ma_{2^{m+1}}.$$ (ii) Use this to show that: ...
0
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0answers
20 views

Calculus of convergence of a series.

During solving a stochastic problem I've reached the following series: $\sum_{n=2, n \in Even}^{\infty} n*\frac{\lfloor2^n/3\rfloor}{2^n}$ but I have no idea how to find the convergence of this ...
3
votes
1answer
80 views

Is series $\sum_{n=0}^{\infty} \Big(\frac{x}{1-x}\Big)^n \frac1{(1-x)^2}$ uniformly convergent for $x <\frac1{2}$?

I believe by the ratio test that it converges for $x < \frac1{2}$ but I can't seem to apply a Weierstrauss M test or other test to show uniform convergence. Maybe it is not.
0
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1answer
13 views

Big O of a difference

Assume $f,g$ are such that $$\lim\limits_{n\to\infty}\frac{f(n)}{g(n)}=r\in\mathbb{R}.$$ Is there anything non-trivial we can infer about $$\left|\frac{f}{g}-r\right|$$ in terms of big-O notation, ...
2
votes
1answer
28 views

Proving that an alternating sequence does not converge

I have the sequence $$(a_n)_{n \in {\mathbb{N}}} = \lim_{n\to\infty} \frac{\frac{n!}{n^n}+1}{\frac{n!}{n^n}+(-1)^n}$$ I can see intuitively why this doesn't converge as it acts like $(-1)^n$ for large ...
1
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1answer
17 views

Convergence in SOT and norm boundedness in $C_r^*(S_\infty)$ equivalent to norm convergence

Let $S_\infty$ - permutation group of the natural numbers fixing all but a finite number of element. And let $C_r^*(S_\infty)$ - reduced group $C^*$-algebra that acts on $\ell_2(S_\infty)$ in obvious ...
4
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0answers
117 views

Convergence of an implicitly defined sequence of random variables

Let $\{X_n\}_{n\ge 1}$ be a sequence of independent identically distributed Poisson random variables with mean $\lambda^*$. Consider a sequence of random variables $\{\hat{\lambda}_{n}\}_{n\ge 1}$ ...
0
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1answer
48 views

$\ell_2$ convergence and $\ell_1$ norm convergence implies $\ell_1$ convergence

Let $x_n \in \ell_2$ converge to $x_\infty \in \ell_2$ and $||x_n||_1$ converge to $||x_\infty||_1$ where $||\cdot||_1$ is $\ell_1$ norm. Is it true, that $x_n$ converge to $x_\infty$ in $\ell_1$?
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1answer
24 views

Convergence/divergence of $\int_0^1 \frac{\sin(\frac{1}{x})}{\sqrt{x}}dx$?

Does the following converge or diverge: $$\int_0^1 \frac{\sin(\frac{1}{x})}{\sqrt{x}}dx$$ I was thinking simply $$\int_0^1 \bigg |\frac{\sin(\frac{1}{x})}{\sqrt{x}}dx \bigg | \le \int_0^1 ...
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0answers
20 views

Convergence of $\prod\limits_{t\in\mathbb{N}}P(|X_t|<\lambda^t)$, where $\lambda>1$

Let $\{X_t\}_{t\in\mathbb{N}}$ be an independent sequence of continuous random variables on the real line. Let $\lambda>1$. I am interested in the quantity $$ P(\forall t\in\mathbb{N},\,|X_t| \le ...
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0answers
43 views

Compute the radius of convergence and interval of convergence of $\sum_{n=1}^\infty\left (\frac {4+2(-1)^n} 5\right)^nx^n $

Compute the radius and interval of convergence of: $$\sum_{n=1}^\infty \left(\frac {4+2(-1)^n} 5\right)^nx^n .$$ I went about this question by applying the root test and this is what I have gotten ...
2
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1answer
20 views

Confusion regarding Gibbs phenomenon

I learned that the partial sum of the fourier series at a jump discontinuity always overshoots the value of the original function by about 9% and this percentage does not die out as we increase the ...
2
votes
1answer
40 views

Why the sum of two divergent integrals has to be divergent?

Suppose $f(x)$ is a function defined on $\mathbb{R}\setminus\{c\}$, where $c$ is a scalar. Consider the integral $$\int_a^bf(x)dx,$$ where $a$ and $b$ are such that $a<c<b$. All Calculus books ...
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0answers
36 views

Does the series $\sum\limits_{n=1}^\infty(-1)^{n+1}\frac{2(n+1)^{3/2}}{n^{3/2}+1}$ converge?

I'm trying to prove that the series $$\sum\limits_{n=1}^\infty(-1)^{n+1}\frac{2(n+1)^{3/2}}{n^{3/2}+1}$$ actually converges. However, $$\lim_{n\to\infty}\frac{2(n+1)^{3/2}}{n^{3/2}+1}=2\neq0$$then the ...
0
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1answer
17 views

Determine the convergence of integral-Bound help needed

I have the following intergal: integral from 0 to infinity of (x^2)/(2x^3-x+1). I do not know how to create an inequality that will help me determine this convergence. Also I have a general question: ...
0
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1answer
29 views

Does the alternating series converge?

I'm trying to find out whether the series $$\sum\limits_{n=1}^{\infty}(-1)^n\ln\left[\frac{8n+2}{7n+1}\right]$$ converges or not, but the alternating series test seems not to apply. What other tests ...
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1answer
41 views

Trigonometric series convergence

I was interested in evaluating $$ \sum_{n=1}^{\infty} \frac{\cos n}{n+k} $$ I saw with computation that many of such series converge. Is there a general result? I've tried using Taylor expansion of ...
1
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1answer
23 views

Prove that the sequence $[f_n]_{n \in N}$, $f_n(x)=\frac{x^{2n}}{1+x^{2n}}$ converges uniformly on $x \in [1+\delta,\infty[$

I've been asked to prove that the sequence $[f_n]_{n \in N}$ with $f_{n}(x)=\frac{x^{2n}}{1+x^{2n}}$ converges uniformly on $x \in [1+\delta,\infty[$ where $\delta > 0 $. So far I've found that ...
0
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1answer
17 views

Distribution of the radius of a circle uniformly on a square

Suppose U1, U2, . . . , V1, V2, . . . are independent Uniform(1, 1) random variables, so that (U1, V1), (U2, V2), . . . are independent points distributed uniformly over the square $[-1,1]^2$. Let ...
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1answer
15 views

Verify whether continuous mapping theorem is applicable

Consider we have $X_n \overset{a.s.}{\to} x$ where $X_n = [A_n, B_n, C_n]$ are $\mathbb{R}^3$-valued random variables, for all $n$ $A_n,B_n,C_n > 0$ almost surely, $A_n,B_n,C_n$ may be dependent or ...
2
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1answer
36 views

Prove that the series $\sum_{1}^{\infty}a^n \frac{logn}{n^2}$ is convergent only for $[-1,1]$ and divergent elsewhere.

Prove that the series $\sum_{1}^{\infty}a^n \frac{logn}{n^2}$ is convergent only for $a \epsilon [-1,1]$and divergent elsewhere. I have given it an honest attempt, I can see why this must be true ...
1
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0answers
34 views

Theorem 3.22 in Baby Rudin: Is this proof correct?

Here's Theorem 3.22 in the book Principles of Mathematical Analysis by Walter Rudin, third edition. $\sum a_n$ converges if and only if for every $\varepsilon > 0$ there is an integer $N$ such ...
0
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0answers
21 views

Prove or disprove that $\int {f^\alpha} \underset{\alpha\rightarrow 1+}{\rightarrow} \int f$

Let $A\subseteq \mathbb{R}$ be a measurable subset with a finite measure Let $f:A\rightarrow \mathbb{R}$ so that $f\in L^1(A)$ be a non-negative function Prove or disprove that: $\underset{\alpha ...
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1answer
36 views

Proving a Limit using the Definition of Convergence

Find $\lim_{n \to \infty} \sqrt \frac{n-1}{n}$ using the definition of convergence. Hi, I am struggling to pick an $\epsilon$ to solve this problem, I know the limit converges to 1, but I am stuck at ...
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0answers
40 views

Does the sequence converge to $b$ with $0<b<a$? [closed]

Let $(a_n)$ and $(b_n)$ be two positive monotonously increasing sequences and $b_n<a_n$. Moreoever, let $\lim_{n}\frac{1}{n}\log(a_n)=a >0$. Does this imply that $$ ...
0
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1answer
28 views

Alt Series convergence or divergence

$$\sum_{k=1}^\infty \frac{(-1)^n n^2}{(n+1)^2}$$ The ratio test came up inconclusive, and I don't know where to go from there.
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1answer
39 views

Convergence of coeffiecients of element in infinite symmetric group algebra

Let $\mathbb{C}S_\infty$ - infinite symmetric group algebra (generated by all finite permutations, i.e. $\mathbb{C} S_\infty = \bigcup_{n=1}^\infty \mathbb{C} S_n$). Turn it into $C^*$-algebra by ...
2
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1answer
34 views

Show that if $f_n$ converges uniformly to $f$, and $f$ has $k$ zeroes. Then $f_n$ has $k$ zeroes when $n>N$

The question Let $(f_n)_{n≥1} ∈ \mathcal{H}(Ω)$ such that $\ f_n \overset{Ω}{\implies} f$. Show that if $f$ has $k$ zeros (counting multiplicities) at $Ω$, then for a $N≥1$ the function $f_n$ has at ...
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2answers
54 views

How do I show that $\ln(2) = \sum \limits_{n=1}^\infty \frac{1}{n2^n}$?

My task is this: Show that $$\ln(2) = \sum \limits_{n=1}^\infty \frac{1}{n2^n}$$ My work so far: If we approximate $\ln(x)$ around $x = 1$, we get: $\ln(x) = (x-1) - \frac{(x-1)^2}{2} + ...
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1answer
20 views

On computations around $\sum_{n=1}^N\frac{n\Lambda(n)}{n+N}$, where $\Lambda(n)$ is von Mangoldt function

By specialization with $F(x)=\frac{1}{1+x}$ in Apostol's Theorem 4.17 (Apostol, Introduction to Analytic Number Theory (Springer)), for intergers $N\geq 1$ one has $$\frac{\log ...
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4answers
84 views

Prove the infinite sum $\sum_{k=0}^{\infty}{\frac{(2k-1)!!}{(2k)!!}x^{2k}}=\frac{1}{\sqrt{1-x^2}}$

I have been trying to prove that $$\sum_{k=0}^{\infty}{\frac{(2k-1)!!}{(2k)!!}x^{2k}}=\frac{1}{\sqrt{1-x^2}}$$ I have done it by using the binomial formula, bit we can't use the gamma function and ...
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0answers
27 views

Therem 3.17 in Baby Rudin: The Analogous Result

Here's Theorem 3.17 in the book Principles of Mathematical Analysis by Walter Rudin, third edition. Let $\{s_n\}$ be a sequence of real numbers. Let $E$ denote the set of all the subsequential ...
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0answers
16 views

Show the sample mean converges to minus infinity when ${X_n}$ are i.i.d. and $E(X_n^+)<\infty$ and $E(X_n{^-})=\infty$

Suppose ${X_n}$ are iid and $E(X_n^+)<\infty$ and $E(X_n{^-})=\infty$. Show that if $S_n := \sum_{j=1}^nX_n$ then $\frac{S_n}{n} \rightarrow -\infty$ almost surely. A hint in my book says to use ...
-1
votes
3answers
37 views

Does the series diverge or converge and find the sum if possible. [closed]

Does the series diverge or converge? $$\sum_{n=1}^\infty \frac{3}{5^n - e^n}$$
0
votes
1answer
35 views

Series Diverge or Converge

Do these series coverage or diverge? What test would you use to show this? Find the sum of the series when possible. I am stuck with this one and I don't know how to go about it. $$\sum_{i=1}^\infty ...
2
votes
2answers
40 views

Weak and strong convergence in $L^p$

Another practice qual question: Let $X = [-\pi,\pi]$ and consider the Lebesgue measure. Let $p$ be a real number with $1 \leq p < \infty$. Define for each integer $k \geq 1$ that $f_k(x) = ...
0
votes
0answers
19 views

Convergence of sum of reciprocal of powers of linear combinations of integrs

From calculus comparison theorems, we have the following well-known result: $\displaystyle\sum_{n\ge1} \dfrac{1}{n^k}$ is absolutely convergent iff $k>1$, where k is a real number. I'm currently ...
1
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0answers
30 views

Theorem 3.55 in Baby Rudin: Every re-arrangement of an absolutely convergent series converges to the same sum in every normed space?

Here's Theorem 3.55 in the book Principles of Mathematical Analysis by Walter Rudin, third edition. If $\sum a_n$ is a series of complex numbers which converges absolutely, then every ...
1
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0answers
24 views

Understanding a theorem about double series

In baby Rudin, there is a theorem stated that: Given a double series $\{a_{ij}\},i=1,2,3,...,j=1,2,3,...,$ suppose that $$\sum_{j=1}^{\infty}|a_{ij}|=b_{i}$$ and $\sum b_i$ converges. Then ...