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

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3
votes
1answer
14 views

Finding a generalized form for this series

While i was just playing around with series i came across this one, $$ S = \sum_{k=1}^\infty[\frac{k}{k-\frac{1}{2}}+\frac{k-\frac{1}{2}}{k}-\frac{k+\frac{1}{2}}{k} - \frac{k}{k-\frac{1}{2}}] $$ ...
1
vote
2answers
37 views

Proving Polynomial is Analytic

If a function $f$ at $x = a$ equals it's Taylor Series, $f$ is said to be analytic. So, if I were given a polynomial $p(x) = \sum_{n=0}^{200}{a_nx^n}$, and trying to prove that $p(x)$ was analytic ...
3
votes
1answer
46 views

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

Let $(a_n)$ be a convergent sequence with $\lim\limits_{n\to \infty} a_n = a$. Show that $$\lim_{n\to \infty} |a_n| = |a|$$ Then state and disprove the converse statement. In order to prove ...
3
votes
2answers
42 views

Uniform convergence of $\sum f(x)^n$

Let $f:X\to\mathbb{R}$ be such that $\sup\{|f(x)|:x\in X\}<1.$ Show that $\sum_{n=1}^{\infty} f(x)^n$ converges and compute the sum.. Every value given by $f$ is less than one, then if ...
1
vote
1answer
20 views

Understanding a step in a double series proof

I'm really confused, how do they get from the first line to the second line ? $$\begin{align*} ...
2
votes
2answers
51 views

If $(1+x+x^2)^{25} =\sum^{50}_{r=0} a_r x^r$ then …

If $$(1+x+x^2)^{25} =\sum^{50}_{r=0} a_r x^r$$ then find : $\sum^{16}_{r=0} a_{3r} =$ My approach : let (1+x) =t therefore, $(1+x+x^2)^{25} =\sum^{50}_{r=0} a_r x^r$ =$(1+x+x^2)^{25} = ...
2
votes
1answer
49 views

Sequences for that $\sum_{n} \frac{1}{x_n}$ is divergent and $\sum_{n} \frac{1}{x_n \ln x_n}$ is convergent

We will denote with $(x_n)$ a given sequence and we introduce the following two series. $$S^* = \sum_{n} \frac{1}{x_n} \quad \text{and} \quad S_* = \sum_{n} \frac{1}{x_n \ln x_n}.$$ We know that if ...
17
votes
3answers
533 views

Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.

What is the closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$? We can use Fourier series of $e^{-bx}$ ($|x|<\pi$) to evaluate $\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}$. But this ...
1
vote
1answer
7 views

Show that $4 - Un+1 < 1/2(4 - Un)$

Let Un be a sequence such that : U0 = $0$ ; Un+1 = $sqrt(3Un + 4)$ We know (from a previous question) that Un is an increasing sequence and Un < $4$ Show that $4$ - Un+1 <(or =) 1/2(4-Un) I ...
3
votes
1answer
31 views

Absolute convergence of $ \sum a_nx_n $ implies absolute convergence of $ \sum a_n$

I'm trying to find a proof (or a conunter example, but I'm somehow convinced that the statement is true) for the following fact: $$ \forall_{(x_n)_{n=1}^{\infty} \lim{x_n} = 0 } ...
2
votes
1answer
28 views

Series similar to harmonic function [on hold]

How to prove that series $\sum_{n=1}^{\infty}{(-1)^{\lfloor{\sqrt{n}\rfloor}}\frac{1}{n}}$ converges?
12
votes
3answers
124 views

An equivalent for $\sum_{n=0}^{\infty} e^{-x\sqrt{n}}$ as $x$ tends to $0^+$

I would like to obtain an equivalent form for $$ f(x)=\sum_{n=0}^{\infty} e^{-x\sqrt{n}} $$ as $x \rightarrow 0^+$. I tried without success to "remove" the $\sqrt{\cdot}$ in the summand by summing ...
0
votes
2answers
22 views

What is the value of the following summation?

Compute $$\displaystyle\sum \limits_{n=0}^\infty (-1)^{n+1} \frac{1}{9^n(2n+2)}$$ I am given the fact that $$ \frac{1}{2}\ln(1+x^2) = \sum \limits_{n=0}^\infty (-1)^n\frac{x^{2n+2}}{2n+2} $$ ...
1
vote
0answers
29 views

Closed-form expressions of $\sum_{n=1}^\infty \frac{\sin^2(an) e^{-bn^2}}{n^2}$

Does anybody know if there's a closed-form expression of this series? $$\sum_{n=1}^\infty \frac{\sin^2(an) e^{-bn^2}}{n^2}$$ where $a$ and $b$ are strictly positive. It's easy to see that it's ...
1
vote
4answers
87 views

Infinite sum of products on my mid-term

This problem was on my calculus mid-term : Determine the convergence or divergence of the series $$ \sum\limits_{n = 1}^\infty {\prod\limits_{k = 1}^n {\frac{{4k - 3}}{{4k - 1}}} } $$ I tried ...
1
vote
0answers
34 views

Is there a name for expressions of the form $\sum_{i=1}^N \prod_{k=1}^i a_k$?

Is there a name for expressions of the form $\sum_{i=1}^N \prod_{k=1}^i a_k$? If so, where can I find the equivalent of a Wikipedia entry?
0
votes
3answers
61 views

Sequence Notation in Analysis

If real sequence is a function from the set $\mathbb N$ to the set $\mathbb R$ and function is represented by $(a,b)$, where $a$ is domain and $b$ is range, then why do we represent sequence only by ...
0
votes
1answer
17 views

Recurrence Relation, a question about the relation between $A_n$ and $A_{n+1}$

Given the following recurrence relation: $$A_{n+1} = {1 \over 4(1-A_n) }$$ $$ A_1 = 0 $$ Can I safely assume that if- $$\forall n \in \mathbb{N}, \ A_n < 1/2$$ then $$\forall n \in \mathbb{N}, \ ...
0
votes
0answers
15 views

Rational Binomial for Taylor series

I want to write this as a sum while $x_0 = 0$ $$f(x) = e^{-x}(1-x)^{-1/2}$$ I know the sum for $e^{-x}$ but I can't figure out the sum for $(1-x)^{-1/2}$ What I tried (minuses cancel out because of ...
0
votes
2answers
21 views

Prove that Recurvisv limits are equal [on hold]

Prove that: $\lim_{n \to \infty} a_n = \lim_{n \to \infty} a_{n-1}$ Ideas?!?
1
vote
1answer
31 views

Convergence of the sequence $f_n(x)=\frac{1}{1+nx^2}$

I'm trying to find the convergence of $f_n$ and $f_n'$ where $f_n(x)=\frac{1}{1+nx^2}$. From the function if I derivate the result is $f'n= -\frac{2nx}{(1+nx^2)^2}$. To determine $f$ I have to take ...
1
vote
2answers
43 views

What is the limit of this sequence?

Problem 3 in the Exercises after Chapter 3 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $s_1 \colon= \sqrt{2}$, and let $$s_{n+1} \colon= \sqrt{2+\sqrt{s_n}} \mbox{ for ...
2
votes
3answers
96 views

If a set $S\subset\mathbb R$ is not closed, does it contain a convergent sequence with a limit outside of $S$?

Suppose S is a subset of $\mathbb{R}$ and that S is not closed. Must it follow that there is a convergent sequence in S that converges to some l not in S?
2
votes
1answer
17 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[ ...
3
votes
5answers
53 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 $2xe^x$. ...
1
vote
1answer
18 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
3answers
80 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
0answers
31 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 ...
2
votes
4answers
104 views

Is $\sum\frac{1}{\sqrt{n+1}}$ convergent or divergent?

$$\sum\frac{(-1)^n}{\sqrt{n+1}} \text{and} \sum\frac{1}{\sqrt{n+1}}$$ The first one is an alternating series, so it would just be: $$\sum (-1)^n\frac{1}{\sqrt{n+1}}\Rightarrow ...
2
votes
1answer
24 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 ...
2
votes
1answer
44 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
8 views

Count and description of vertices of certain faces of the Tridiagonal Birkhoff polytope $\Omega^t_{d+k}$

For $k \ge 1$, $d \ge 2$ and $k \le d - 1$, let ${}^f_d\Omega^t_{d+k} (d;c_k(d - 1))$ be the intersection of $k - 1$ facets of the Tridiagonal Birkhoff polytope $\Omega^t_{d+k}$ with equations: ...
1
vote
1answer
34 views

Caratheodory: Alternative Definition

Idea My idea is to facilitate Caratheodory's construction by composing it with Hahn-Kolmogorov. Problem Given a premeasure on a ring $\mu:\mathcal{R}\to\overline{\mathbb{R}}_+$. Do the ...
1
vote
3answers
31 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 $n>m$: ...
1
vote
3answers
47 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 ...
0
votes
1answer
18 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, ...
3
votes
2answers
89 views

Summing the sequence $a(n) = \sin(n x) \exp(-nt)$

Consider the sequence $a(n)$ defined by $a(n) = \sin(n x) \exp(-nt)$, where $n = 0, 1, 2, 3, 4, \ldots$. The parameter $x$ is a real number. Parameter $t$ is a positive real number. It is clear that ...
1
vote
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
19 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
61 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 ...
5
votes
4answers
196 views

How to find the following sum? $\sum\limits_{n = 0}^\infty {\left( {\frac{1}{{4n + 9}} - \frac{1}{{4n + 7}}} \right)} $

I want to calculate the sum with complex analysis (residue) $$ 1 - \frac{1}{7} + \frac{1}{9} - \frac{1}{{15}} + \frac{1}{{17}} - ... $$ $$ 1 + \sum\limits_{n = 0}^\infty {\left( {\frac{1}{{4n + 9}} - ...
1
vote
5answers
41 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 ...
1
vote
0answers
28 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\\ ...
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
80 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
93 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
81 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
1answer
42 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
3answers
47 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 ...
5
votes
2answers
187 views
+50

$\sum_{p \in \mathcal P} \frac1{p\ln p}$ converges or diverges?

We will denote the set of prime numbers with $\mathcal P$. We know that the sum $\sum_{n=1}^{\infty}\frac1n$ and $\sum_{n=2}^{\infty}\frac1{n\ln n}$ diverges. It is also known that $\sum_{p \in ...