For questions about recurrence relations, convergence tests, and identifying sequences

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2
votes
1answer
15 views

Cauchy Product $n$ times

I'm looking for a short and precise proof of the following identity; $$\left(\sum_{k=0}^\infty \frac{C_k}{k!}x^k\right)^n=\sum_{k=0}^\infty\left[ ...
1
vote
1answer
17 views

What is the difference between arithmetic and geometrical series?

What is the difference between arithmetic and geometrical series? Also what are they? How do they look like?
3
votes
5answers
48 views

Taylor Series of $2xe^x$

I have to find the Taylor Series for $2xe^x$ centred at $x=1$. I came up with the following. $$e^x = e^{x-1} \times e = e \bigg( \sum_{n=0}^\infty \frac{(x-1)^n}{n!}\bigg)$$ Then consider $2xe^x$. ...
2
votes
0answers
11 views

How to evaluate the composed euler sum $\sum_{n=1}^\infty \frac{H_{kn}}{n^2}$

Is there a closed form for the following ? $$\sum_{n=1}^\infty \frac{H_{kn}}{n^2}$$ I suspect it is easy for $k={1,2}$ ; but the complexity might increase for greater values Can we generalize ...
3
votes
3answers
76 views

Proving of $\frac{\pi }{24}(\sqrt{6}-\sqrt{2})=\sum_{n=1}^{\infty }\frac{14}{576n^2-576n+95}-\frac{1}{144n^2-144n+35}$

This is a homework for my son, he needs the proving.I tried to solve it by residue theory but I couldn't. $$\frac{\pi }{24}(\sqrt{6}-\sqrt{2})=\sum_{n=1}^{\infty ...
2
votes
1answer
30 views

Show that $\lim_{n\to ∞} |a_n| = |a|$ if $a_n\to a$

Let $(a_n)$ be a convergent sequence with $$\lim_{n\to ∞} a_n = a$$. Show that $$\lim_{n\to ∞} |a_n| = |a|$$ Then state and disprove the converse statement. In order to prove that I would use the ...
2
votes
1answer
22 views

How to use generating functions to partially sum multiple integer sequences?

Let's say I want to find the following double sum $$ \sum_{k=1}^mk\sum_{n=1}^kn={1\over24}m(1+m)(2+m)(1+3m) $$ but using a generating function for the involved sums. The polynomial generating function ...
1
vote
3answers
28 views

Prove a sequence is Cauchy and find its limit

Let $a\leq b \in \mathbb{R}$. Show that the sequence $a_1 = a, a_2=b$ and $a_{n+2}=\frac{a_{n+1}+a_n}{2}$ for $n\geq 1$ is Cauchy and find it's limit. I did for $n>m$: ...
2
votes
1answer
37 views

What are the connections between the three Mertens' theorem?

In number theory the three Mertens' theorems are the following. Mertens' $1$st theorem. For all $n\geq2$ $$\left\lvert\sum_{p\leqslant n} \frac{\ln p}{p} - \ln n\right\rvert \leq 2.$$ Mertens' ...
0
votes
1answer
14 views

Lowest divisible number in number string

A number is arranged in a pattern like: 12345678910111213141516... What is the lowest value of that pattern divisible by 72? They are single numbers, not seperate (i.e. first in sequence is 1, ...
1
vote
3answers
45 views

Two questions on number 2013

a) All natural numbers from $1$ to $2013$ are written in a row in an order. Can you insert '+' and '-' signs between them so that the value of the resulting expression is zero? If it is so how many ...
1
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1answer
25 views

Define sequence and convergence

Define function f: $\mathbb{R}_+\rightarrow\mathbb{R} $ by: $ f(x)=\sqrt{\frac{x^2}{3}+\frac{18}{x}}$ 1) Show that $f'$ has one minimum/maximum, define $f'$s monotony conditions and sketch $f$. I ...
0
votes
1answer
17 views

Convergence of the sequence of maxima of a function sequence

Suppose we have a compact set $K \subset \mathbb{R}$ and a sequence of continuous functions $f_n: K \rightarrow \mathbb{R}$. Let $f$ be the uniform (and hence continuous) limit of $(f_n)_n$. Assume ...
2
votes
2answers
58 views

Show that a Series Diverges

Question: Let a sequence ($a_n$) have the property $\lim \limits_{n \to \infty} na_n = a > 0$ Show that the series $\sum_{n=1}^\infty a_n$ diverges Attempts: Basically, I firstly tried ...
1
vote
5answers
40 views

Prove Using Definition that the Sequence $\frac{n+1}{\sqrt{n}+2}$ Diverges to Infinity

Formally, a sequence $x_n$ diverges to infinity whenever for all $M>0$ there exists $N(M)$ such that $n>N(M)$ implies $x_n>M$. Prove formally, using the definition, that the following ...
3
votes
1answer
44 views

Prove $\cos x = \frac{8}{\pi}\sum_n \frac{n\sin 2nx}{4n^2-1}$ with Fourier series

I want to prove $$\cos x = \frac{8}{\pi}\sum_n \frac{n\sin 2nx}{4n^2-1}\;x\in(0,2\pi)\;\;\;\;[1]$$ I have two questions regarding this: $(1)$ How can I find a function $f$ such that the former ...
-1
votes
1answer
79 views

Proof that it is not uniformly convergent on R [on hold]

Prove that the series $$\sum_{n=1}^\infty 2^n \sin \left( \frac{x}{3^n} \right)$$ is not uniformly convergent on R.
8
votes
1answer
91 views

Evaluate $\sum\limits_{n=1}^{\infty} \frac{2^n}{1+2^{2^n}}$

How to evaluate the infinite series: $$\sum\limits_{n=1}^{\infty} \frac{2^n}{1+2^{2^n}}$$
3
votes
2answers
78 views

$\frac{1}{x^2} \int xe^x dx$ without using integration by parts

On a test i just had, i needed to solve a differential equation which lead me to having to find the result of $$ \frac{1}{x^2}\int xe^x dx $$ I then attempted to do this integral without integration ...
1
vote
3answers
44 views

About the infinite geometrical sequence factored with n [duplicate]

I just came across this thread, and i asked myself: I know that $\sum^\infty_{n=0} x^n = \frac{1}{1-x}$ But what happens when we set up the sum like $$\sum^\infty_{n=0} nx^n = ?$$ There is ...
1
vote
0answers
22 views

Series in closed form

Let $\alpha \in (0,1)$ and $z\geq 0$. Define the function: $$ \theta_{\alpha}(z)=\sum_{j=0}^{\infty}\left(\frac{z}{\alpha j + 1}\right)^j. $$ Can $\theta_{\alpha}(z)$ be written in closed form? Are ...
1
vote
1answer
40 views

The asymptotic behavior of $\sum_{n=1}^\infty\frac{1-\cos(x4^n)}{2^n}$ as $x\to 0$

Is there a way to show that for small $x$'s $$\sum_{n=1}^\infty\frac{1-\cos(x4^n)}{2^n}\le c\sqrt x$$ I tried Taylor expansion of $\cos$ and square root... Thank's
1
vote
1answer
27 views

How to prove matrix geometric convergence to any matrix?

Suppose I have two vectors $x$ and $v$, and we want to calculate the following expression: $$(I+x\cdot v^{T})^{-1}$$ My professor affirmed that we could treat this as a "geometric progression" ...
0
votes
2answers
28 views

How to check convergence of the following series

How to check convergence of: $1.\sum_{n=1}^\infty \frac{(n+1)^n}{n^{n+\frac{5}{4}}}$.I tried using Cauchy's root test but got limit=1.How to do it? $2.\sum_{n=1}^\infty \frac{1}{n^{1/2}}tan ...
0
votes
0answers
17 views

When does the limit of the ratio of consecutive terms of a sequence exist?

I am trying to understand and obtain some sufficient conditions under which the limit of the ratio of consecutive terms of a sequence exists. Let $x_n$ be a sequence of positive integers, such that ...
0
votes
1answer
34 views

Convergent squence in topology

Please, I consider this topological space $(E,\tau)$ where $\tau=\{G\subset E, ~\text{card}~ (E\setminus G)~\text{countable}\}\cup\{\emptyset\}$ How to prove that a sequence $(x_n)$ is convergent in ...
0
votes
1answer
17 views

How is OEIS sequence A120933 'maximal leading nondecreasing subword ' to be understood?

For n=2 we only have these four binary words: 00 01 10 11 What is the procedure for calculating by hand T(2,1) and T(2,2)? I'm trying to understand the reasoning behind this sequence as I can't see ...
5
votes
2answers
85 views

Evaluation of $\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$

During my calculation I ended with the following product: $$P=\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$$ I tried to express in term of series by taking the logarithm ...
1
vote
1answer
38 views

Infinite series and the Riemann zeta function

I have two questions concerning infinite series in the context of the Riemann zeta function. Given the properties of infinite series, why can't we regroup the terms in $\zeta(0)$ in such a way as to ...
4
votes
2answers
73 views

prove the limit of $k^{1/k}$ is $1$ [duplicate]

I want to prove that the limit of the sequence $k^{1/k}$ is $1$ as $k$ tends to infinity without using advanced rules such as L'Hospital's Rule and just using the basic rules in real analysis. How ...
1
vote
0answers
25 views

finding a function whose values at certain points are defined

I have certain polynomials in a sequence from which I am trying to derive the common term. The polynomials are: $7k^2-8\\ 21k^4-368k^2-704\\ ...
0
votes
1answer
23 views

Sequence word problem [on hold]

A basketball is dropped from 81 meters atop the tower. If it rebounds up 2/3 of the distance after each bounce, what is the total vertical distance traveled by the ball before it come to rest?
3
votes
1answer
21 views

Convergence of monotone $f_n:[0, \infty) \rightarrow [0,1]$ to continuous, monotone $g$ is uniform

Let $g: [0, \infty) \rightarrow [0,1]$ be a continuous, monotone increasing function where $g(0)=0$ and $g(x)\rightarrow 1$ as $x \rightarrow \infty$. Also let $f_n:[0, \infty) \rightarrow [0,1]$ be a ...
2
votes
1answer
22 views

For $s_n$ a sequence in $\Bbb R$, if $\lim s_n$ defined as a real number, then $\liminf s_n = \lim s_n = \limsup s_n$.

I'm studying some real analysis, and I'm trying to figure out how to prove the following theorem. For the most part, I've got things figured out. I'll post the theorem and my proof (which closely ...
1
vote
2answers
154 views

Finding Sum for Infinite Series

Normally when I keep try multiple ways to solve a problem, I get an idea of where to start, and eventually can solve it. But it hasn't been the way for this question and I've been stuck for hours. ...
0
votes
0answers
11 views

Tips on effectively representing this recurrence relation in a generalized form.

The recurrence relation is $$ y_n = d_1 y_{n-1}+\frac{d}{dx}[y_{n-1}] $$ a good thing to note as well is $$ \frac{d}{dx}[d_n] = d_{n+1} $$ This is terrible to expand out after a good while, The main ...
0
votes
3answers
32 views

Prove that $s_n\leq 2-\frac{1}{n}$ for all $n\geq 1$

We have the sequence $(s_n)_{n\geq 1}$ given by $s_n=\sum^n_{k=1}\frac{1}{k^2}$. Prove that $s_n\leq 2-\frac{1}{n}$ for all $n\geq 1$. Thanks in advance!
5
votes
1answer
53 views

Proof that the harmonic series is < $\infty$ for a special set..

In one of my books i found a very interesting task, i am really curios about the solution: Let $M = \{2,3,4,5,6,7,8,9,20,22,...\} \subseteq \mathbb{N}$ be a set that contains all natural numbers, ...
1
vote
1answer
66 views

Give an example where $\{a^2_n\}_{n=1}^\infty$ is a Cauchy sequence, but $\{a_n\}_{n=1}^\infty$is not.

Give an example where $\{a^2_n\}_{n=1}^\infty$ is a Cauchy sequence, but $\{a_n\}_{n=1}^\infty$is not. Could somebody give me an example of this? Thanks in advance!
0
votes
0answers
29 views

Convergence of $|f_n(x+n)-f(x+n)| \le \frac{\log\log \left(1+\frac{x+n}{\gamma}\right)}{\log(1+n)}$ on $x \in [0,1]$

I have the following relationship between $f_n$ and $f$ on $x \in [0,1]$ \begin{align*} |f_n(x+n)-f(x+n)| \le \frac{\log\log \left(1+\frac{x+n}{\gamma}\right)}{\log(1+n)} \end{align*} for all $x \in ...
-1
votes
0answers
19 views

Series comparison test for $ \sum_1^\infty n^{\ln(n)} \ln(n^n)$

$$ \sum_{n=1}^\infty n^{\ln(n)} \ln(n^n)$$ Which function should I use to compare this to proove that it diverges? To me comparison test for this series the obvious solution.
2
votes
0answers
80 views

Find $\displaystyle\sum_{k=1}^\infty\dfrac{1}{\left(\binom{2014+k}{2014}\right)}$

Find $\displaystyle\sum_{k=1}^\infty\dfrac{1}{\left(\binom{2014+k}{2014}\right)}$, where $\left(\binom{a}{b}\right)=\dfrac{a!!}{b!!(a-b)!!}$ EDIT : Someone pointed out in the Mathematics chat that my ...
1
vote
1answer
26 views

Product of a Convergent Series and Bounded Sequence

Let $a_n$ be a bounded sequence and $\sum_{n=1}^\infty b_n$ be a convergent series. Then $\sum_{n=1}^\infty b_na_n$ is convergent. I have found a counterexample to prove it false; If we let ...
2
votes
2answers
35 views

First five terms of the sequence $a_n=2/e^n$ [on hold]

I just wanted to check if my answer here is correct. I am not sure if I am supposed to simplify (e) even further down, and turn the fraction into a decimal or not because it does mention use ...
2
votes
1answer
35 views

Convergence of the integral of cos(x)/x^2

For $n$ in the natural numbers let $a_n = \int_{1}^{n} \frac{\cos(x)}{x^2} dx$ Prove, for $m ≥ n ≥ 1$ that $|a_m - a_n| ≤ \frac{1}{n}$ and deduce $a_n$ converges. I am totally stuck on how to even ...
2
votes
1answer
16 views

Prove that if $\sum g_k$ converges uniformly on a set $S$ and if $h$ is a bounded function on $S$, then $\sum hg_k$ converges uniformly on $S$

Prove that if $\sum g_k$ converges uniformly on a set $S$ and if $h$ is a bounded function on $S$, then $\sum hg_k$ converges uniformly on $S$ A little confused about this question, would love to ...
0
votes
1answer
37 views

Finding limits superior and inferior of a sequence

Find the limits superior and inferior of the following sequences: note: "For a set, those are the infimum and supremum of the set's limit points, respectively. " $a_n=\frac{n}{n+1} ...
0
votes
1answer
32 views

Weak Convergence and its Relationship to a Sequence of Norms

"Prove that $x_n\xrightarrow{w} x$ implies lim inf$||x_n|| \geq ||x||$." I'm trying to understand weak convergence better through this exercise. Here, $\xrightarrow{w}$ means weakly convergent, i.e. ...
0
votes
1answer
35 views

Functions or sequence divergences link

One can prove that the sequence $ u_n=\{\sin(n)\}_{n \in \mathbb{N}} $ diverges using a similar argument as in : Proves the divergence of sequence of sin(n) But we can also prove that ...
1
vote
1answer
31 views

A property of a sequence

Why if a sequence $a_{n}\rightarrow + \infty$, then it can't be that ${\frac{a_{n+1}}{a_{n}}\rightarrow 0 }$? Thanks!