For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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9
votes
5answers
253 views

Number of points of accumulation of a sequence

Can a sequence have infinitely many points of accumulation i.e. we can extract infinitely many subsequences from it s.t. they all converge to their respective point of accumulation? I have the ...
0
votes
3answers
125 views

A Chinese Exam Question which is…quite hard

Let $f(x)=x^2-2x-3$, and $x_n$ be some sequence. $x_1=2$, $x_n =$ the $x$ coordinate of the point of intersection of the $x$ axis and the line joining $P(4,5)$ and $Q_n(x_n, f(x_n))$. Find an ...
1
vote
3answers
34 views

Where does the following series converge? [on hold]

Using integrals or by any other method find: $\lim_{n \rightarrow\infty} \sum_{i=1}^{n}\frac{1}{n+i}$
-1
votes
0answers
28 views

What is the limit distribution of this sequence of random variables? [closed]

Find the distribution of the limit when $n\to\infty$ of$${S_n\over n^2}$$ where $S_n=X_1+X_2+...+X_n$, and $ X_1,X_2,...$ are random variables i.i.d. with characteristic function ...
9
votes
2answers
190 views

Books about harmonic numbers

I'm looking for books about harmonic numbers, where I could find proofs of results about them. For example a proof for the fact, that the generating function of the generalized harmonic numbers is $$ ...
1
vote
1answer
28 views

Denominators of harmonic numbers: asymptotic behaviour.

About the sequence $d_n$ of the denominators of harmonic numbers, I know these facts: It is unbounded, since $p\mid d_p$ for any prime $p$. It contains only one $1$. What more is known? Specially, ...
1
vote
2answers
29 views

Prove the monotone convergence theorem for sequences of Lebesgue-integrable functions

I'm trying to prove the monotone convergence theorem for $L^1$ functions: Suppose $(f_n)$ is a sequence of $L^1$-functions (i.e Lebesgue-integrable functions) over a measure space $(X,\sigma (X), ...
2
votes
2answers
76 views

Linear approximation to the product: $\prod_{k=0}^r\left(1+\frac12\left(\frac{\frac12+k+1}{\frac12+k}-\frac{\frac12+k}{\frac12+k+1}\right)\right)$

I have come upon with the next expression: \begin{equation} P_r=\prod_{k=0}^r \left(1+\frac{1}{2}\left(\frac{\frac{1}{2}+k+1}{\frac{1}{2}+k} -\frac{\frac{1}{2}+k}{\frac{1}{2}+k+1}\right)\right) ...
1
vote
1answer
25 views

Series with recursive terms: reciprocal of the sums of previous terms

What are some references and material that are devoted to studying series of the form $x_1+\frac1{x_1}+\frac{1}{x_1+\frac1{x_1}}+\dotsc$ where $x_1>0$ is given? For instance, my (non math) friend ...
1
vote
3answers
30 views

Searching for a sequence of functions

Consider the following set of functions: $$ A=\left\{f\in C([0,1],\mathbb{R}): f(0)=0, \lim_{r\searrow 0}\frac{f(r)}{r}\text{ exists}\right\}. $$ Is there a sequence $(f_n)\in A^{\mathbb{N}}$ such ...
0
votes
1answer
528 views

Does the series $\sum_{n=0}^\infty\frac{\sin(n+\frac 12)\pi}{1+\sqrt{n}}$ converge?

Does the series $$\sum_{n=0}^\infty\frac{\sin(n+\frac 12)\pi}{1+\sqrt{n}}$$ This is supposed to be an alternating series but I can't seem to figure out what the $b_n$ is in this case. is there some ...
2
votes
0answers
16 views

$\omega$-limit set of a point $x \in X$

I would like to verify whether the following definition of the $\omega$-limit set of a point $x \in X$ is correct: $$\omega(x) = \{ y \in M : \exists \text{ sequence }\{t_j\}, \text{ where } t_j ...
1
vote
2answers
41 views

How to prove that $\sin x = x \, _{0}F_{1}(-;\frac{3}{2};-\frac{x^{2}}{4})$?

How to prove that $\sin x = x \, _{0}F_{1}(-;\frac{3}{2};-\frac{x^{2}}{4})$? where $_{0}F_{1}$ is the hypergeometric series?
1
vote
2answers
52 views

What is the sum of this series: $1 + \frac{1}{5}x + \frac{1 \times 6}{5 \times 10}x^2 +\cdots$?

Say I have a series like the following; $$1 + \frac{1}{5}x + \frac{1 \times 6}{5 \times 10}x^2 + \frac{1 \times 6 \times 11}{5 \times 10 \times 15}x^3 + \cdots.$$ How do I find the sum of this? ...
-2
votes
1answer
12 views

Averge of sequence converges, sequence bounded? [on hold]

Suppose $\frac{\sum_{k=1}^{n} |a_n|}{n}\leq M$ for every $n\in \mathbb N$,does it imply that $\sup |a_n|<\infty$ ? Can someone give me a hint?
2
votes
5answers
80 views

Example 3.40 (b) in Baby Rudin: How to find $\lim_{n\to\infty} \sup \frac{1}{\sqrt[n]{n!}}$?

Here is Theorem 3.39 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: Given the power seires $\sum c_n z^n$, put $$\alpha = \lim_{n\to\infty}\sup\sqrt[n]{\vert c_n ...
0
votes
3answers
24 views

$\sum(b_j)^\frac{1}{2}\frac{1}{j^a}$ converges

If $b_j>0$ and $\sum b_j$ converges then show that $\sum(b_j)^\frac{1}{2}\frac{1}{j^a}$ converges for any $a>\frac{1}{2}$ My suggestion is that if a is 1/2 or less then you get $\sum ...
0
votes
1answer
76 views

Show that if $(a_k)_{k \in \mathbb{N}}, (b_k)_{k \in \mathbb{N}}$ are bounded, then $\lim \sup(a_k + b_k) \leq \lim \sup(a_k)+\lim \sup (b_k)$ [duplicate]

Show that if $(a_k)_{k \in \mathbb{N}}, (b_k)_{k \in \mathbb{N}}$ are bounded, then $\lim \sup(a_k + b_k) \leq \lim \sup(a_k)+\lim \sup (b_k)$ I'm not quite sure how to prove this. I want to try ...
0
votes
2answers
35 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 ...
1
vote
0answers
36 views

Theorem 3.37 in Baby Rudin: $\lim\inf\frac{c_{n+1}}{c_n}\leq\lim\inf\sqrt[n]{c_n}\leq\lim\sup\sqrt[n]{c_n}\leq \lim\sup\frac{c_{n+1}}{c_n}$

Here's Theorem 3.37 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: For any sequence $\{c_n\}$ of positive numbers, $$\lim_{n\to\infty} \inf \frac{c_{n+1}}{c_n} ...
0
votes
3answers
30 views

Proving the set of subsequences of a sequence are uncountable

I am attempting to solve the following problem. Let ($s_n$) be a subsequence of real numbers. Prove that the set of subsequences of ($s_n$) is uncountable. I was thinking that approaching this ...
2
votes
1answer
89 views

How $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots =\ln 2$? [duplicate]

while doing the Integration problem using Limit of a sum approach i have a doubt how $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots =\ln2$$ by infinite geometric series we have ...
2
votes
1answer
63 views

If $|z_n-z_m|> 2$ for every $n\ne m$ then $\sum \frac{1}{z_n^3}$ converges

Let $(z_n)$ be a sequence of non-zero complex numbers such that $\forall n,m, n\neq m\implies |z_n-z_m|> 2$ Prove that $\sum \frac{1}{z_n^3}$ converges. I'm clueless with this problem. A ...
0
votes
1answer
93 views

Is there any summation method that assigns $ \sum_{n=1}^\infty \frac{1}{n} =-\frac{\pi}{2}$

I don't know too much about alternate summation methods, but am interesting to know if any give the sum of the harmonic series to be $$-\frac{\pi}{2}$$
7
votes
0answers
58 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 ...
1
vote
0answers
45 views

The expected value of the smallest number in sample $S$ is:

We are given a set $X = \{x_1, …. x_n\}$ where $x_i = 2^i$. A sample $S ⊆ X$ is drawn by selecting each $x_i$ independently with probability $p_1 = \frac{1}{2}$. The expected value of the smallest ...
5
votes
3answers
111 views

My formula for sum of consecutive squares series?

I stumbled upon a specific series, who's Sum of squares of consecutive integers equals the sum of squares of the continuation of that consecutive integers. For exmaple, this first number in the ...
0
votes
1answer
33 views

Alternating Euler sums with even index

We are all aware of the generating function of $\frac{x \arctan x}{x^2+1}$ which is: $$\frac{x \arctan x}{x^2+1} = \sum_{m=1}^{\infty} (-1)^m \left ( \mathcal{H}_{2m} - \frac{1}{2} \mathcal{H}_m ...
1
vote
0answers
32 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 ...
1
vote
1answer
24 views

What is the coefficient and constant term in the following sequence defined recursively?

Let $f_n(x)$ be a sequence of polynomials defined inductively as $f_1(x) = (x - 2)^2$ $f_{n+1}(x) = (f_n(x) - 2)^2$ $; n \ge 1$ Let $a_n$ and $b_n$ respectively denote the constant term and the ...
0
votes
2answers
19 views

bounded but not convergent sequences

I am not sure that if this question has a positive answer...I am looking for a sequence of real numbers $(p_{n})_{n\geq 1}$ such that $-1<\lim _{n}\inf p_{n}\leq \lim_{n}\sup p_{n} <1$ (as ...
0
votes
1answer
27 views

Series proof needed

I have following equations but I do not know the proof. Kindly provide the proof or give me some reference to look into. Here are the equations. 1- ...
-2
votes
0answers
26 views

Why is it that when n ≥ 1 the series is $\le$ 1/4 [on hold]

So how is the series $\sum_{n=1}^\infty \frac{1^2 * 3^2 * 5^2 ... (2n-1)^2}{2^2 * 4^2 * 6^2 ... (2n)^2}$ < 1/4 for n $\ge$ 1 is it because the series is divergent outside of the interval of ...
0
votes
0answers
19 views

show function has a uniformly convergent subsequence

alright, so I have this question from my analysis class and I believe I have answered it correctly. I would be grateful if you could read it and give me your thoughts. A sequence $f_n$ of real valued ...
2
votes
1answer
24 views

How to show that all sequentially compact spaces are bounded?

I want to show given a sequentially compact subset $A \subseteq M \implies A$ is bounded. I read this Every sequentially compact set is closed and bounded. but the proof is poorly written and jumpy ...
0
votes
1answer
32 views

Does equality of the sum of two such series imply equality of each term of that series?

Let a(1)< a(2) < ..< a(m) and b(1)< b(2)<..< b(n) be real numbers such that $$\sum_{i=1}^m |a(i)-x| = \sum_{j=1}^n |b(j)-x|$$ for all x belonging to R. Show that m=n and ...
0
votes
0answers
34 views

When adding or subtracting two infinite sums, why is there no issue with “staggering” or arbitrarily manipulating the “alignment” of terms?

I was watching Ramanujan: Making sense of 1+2+3+... = -1/12, where the presenter writes: (I tried to write this out in $\LaTeX$ but couldn't figure out how to do multi-column alignment without ...
0
votes
2answers
568 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 ...
0
votes
1answer
21 views

Sum of product of convergent series

Suppose $\{a_n\},\ \{c_n\}\subset\mathbb{R}$ satisfies $\lim\limits_{n\to\infty} a_n = a\in \mathbb{R}$, $\lim\limits_{n\to\infty} c_n = 1$. Prove that $$ \lim\limits_{M\to\infty} ...
0
votes
1answer
23 views

Proof of the Strenghtened Limit Comparison Test

I'm studying on my own using Bonar and Khoury's Real Infinite Series. I understand the proof of the "regular" Limit Comparison Test( a link to google books, page 23 ) but the book doesn't provide a ...
0
votes
2answers
46 views

$\sum_{k=1}^n 1/k - \log n$

I got this question : $$a_n = \sum_{k=1}^n \frac 1k - \log n$$ I proved that $\lim a_n $ exist. Now I have to prove: $$ 0<a_n-\lim a_n\le \frac 1n $$ for every $n \in \mathbb N$. I tried ...
4
votes
2answers
137 views

A result on sequences: $x_n\to x$ implies $\frac{x_1+\dots+x_n}n\to x$ without using Stolz-Cesaro

If $x_n \to x$, how might we prove $$\lim_{n \to \infty} \frac{\sum_{i=1}^{n} x_i}{n} = x$$ Of course, one has $\limsup x_n = \liminf x_n = x$, and thus, using the Stolz-Cesaro theorem: $$\liminf ...
1
vote
1answer
18 views

Is this an incorrect error bound value?

In Step 3, they are determining the $(n+1)^{th}$ term. I think the proofreader just added 1, instead of subbing in (n+1). Is that right? I think the correct term should be ...
10
votes
1answer
274 views

Limit of recursive sequence $n^2q_n=1+(n-1)^2q_{n-1}+2(n-2)q_{n-2}$

When looking at this riddle, I came across the following sequence for the frequency of sampled integers between 1 and $n$ in a without replacement/without neighbour sampling: $$q_1=1,\quad ...
0
votes
1answer
21 views

is this an upper bound for the square of a finite series?

I'm trying to show that the derivative of a (long) function is negative. after a lot of simplifying, i need the following inequality to hold: $\displaystyle \Big ( \sum_{i=1}^k n_i \Big )^2 < ...
-1
votes
2answers
42 views

$\sum \frac{b_j}{j^2}$ converge or diverge

if $b_j>0$ and $\sum b_j$ convege does $\sum \frac{b_j}{j^2}$ converge or diverge I think it converges because $\sum \frac{b_j}{j^2} < \sum b_j$ Thus by comparison test converges.
114
votes
1answer
4k views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where ...
0
votes
1answer
36 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 ...
0
votes
1answer
50 views

If $A_n = \sum_{k = 1}^n \dfrac{k^6}{2^k}$, find $\lim_{n \to \infty} A_n$

Set $$A_n = \sum_{k = 1}^n \dfrac{k^6}{2^k}.$$ Find $\displaystyle \lim_{n \to \infty} A_n$. I tried solving this using a reduction method. That is, reducing the above series to an ...
0
votes
1answer
26 views

How would I show $|x| \le 1$ given the equation for $x$ the expression in the equation?

The expression is $x = \sin(\theta /2)$. I am asking how would I show that $\sin(\theta/2)\le1$ based on the expression? I already know that the biggest $\sin$ will ever get is $[-1, 1]$ which is the ...