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

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0
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1answer
8 views

Sequence of quadratic polynomials

Let $P_n$ be a sequence of Quadratic polynomials on $[0,1]$ such that $\lim_{n \rightarrow \infty}P_n(a_i) = b_i$ for $i = 1,2,3$ where $b_i$ are real numbers. Then 1) $P_n$ converges pointwise in ...
4
votes
0answers
41 views

Uncountable series without axiom of choice

Consider a sequence of positive real numbers $(\alpha_i)_{i\in I}$ for some (suppose maybe wellordered for now) set $I$. Using axiom of choice, it is easy to see that $\sum_i \alpha_i$ is infinite if ...
0
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0answers
14 views

Taylor Polynomials: Estimating accuracy of an approximation f(x) ≈ Tn(x)

$f(x) = \sqrt{x},\space\space\space\space a = 4,\space\space\space\space n = 2,\space\space\space\space 4 \le x \le 4.7$ I approximated f by a Taylor polynomial with degree 2 at the number 4. ...
0
votes
1answer
13 views

A basic logical question on on sequence of functions

Let $f_n:[0,\infty)$ and $f:[0,\infty)$ be a sequence of functions such that for every finite $T$, $f_n:[0,T]\rightarrow f:[0,T]$ uniformly. But, it need not be true that $f_n:[0,\infty)\rightarrow ...
-2
votes
2answers
58 views

How to differentiate a series? [on hold]

I need help with this function. Can this function be differentiated? $\frac{\partial}{\partial b}\ln{L}=\frac{n}{b}+\sum{\ln{x}}-\frac{\partial}{\partial b}(-a^b\sum{x^b})$ I dont know how to last ...
3
votes
2answers
27 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
42 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 ...
2
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1answer
32 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*} ...
1
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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
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1answer
32 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
55 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 ...
2
votes
1answer
30 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?
0
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2answers
23 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
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0answers
30 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
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
1answer
18 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}, \ ...
1
vote
4answers
89 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 ...
3
votes
2answers
60 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} = ...
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
22 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?!?
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
32 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
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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
1answer
18 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
19 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
56 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
34 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
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 ...
3
votes
1answer
47 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 ...
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 ...
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 $n>m$: ...
2
votes
1answer
45 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
19 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
48 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
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 ...
1
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5answers
42 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
81 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
94 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
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 ...
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
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
1answer
28 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
36 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 ...