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

learn more… | top users | synonyms (5)

-2
votes
4answers
47 views

What function can I use to represent the next sequence: 2,2,2,2,2,3,3,2,2,2,2,2,3,3,2,2,2,2,2,3,3… [on hold]

What function can I use to represent the next sequence: 2,2,2,2,2,3,3,2,2,2,2,2,3,3,2,2,2,2,2,3,3... I am looking for general solution.
-3
votes
0answers
21 views

Finding the limit of the series

Find the limit of the sequence , can somebody help ,I'm struck.
1
vote
1answer
14 views

Series Expanding a Function: Complex Answer?

I have $$ f(x)=2\arccos\left(\frac{x}{2}\right)-x\sqrt{1-\frac{x^2}{4}} $$ and my friend says that the series of this function about $x=2$ (truncated to the first term) given $x\leq2$ is ...
1
vote
2answers
48 views

Prove that $\sum_{p \in \mathcal P} \frac1{p\ln \ln p}$ is divergent

In this question I've asked to decide the convergency of the series $\sum_{p \in \mathcal P} \frac1{p\ln p}$. Now I ask you to show that the series $$\sum_{p \in \mathcal P} \frac1{p\ln \ln p}$$ ...
0
votes
0answers
20 views

Continued fraction approximation

Let $\theta\in\Bbb{R}_{\gt0}$. A) Prove that the convergents for the continued fraction expansion of $\theta$ give us better and better rational approximations to $\theta$. B) Suppose $\theta\notin ...
7
votes
1answer
87 views

how to compute this limit

compute $I=\lim\limits_{n\to+\infty}\sqrt[n]{\int\limits_0^1x^{n+1}(1-x)\cdots(1-x^n)dx}$ attempt: i tried to evaluate the integral $$\begin{align} ...
2
votes
2answers
38 views

Generating function for Pell numbers

Problem: The Pell numbers $p_n$ are defined by the recurrence relation \begin{align*} p_{n+1} = 2p_n + p_{n-1} \end{align*} for $n \geq 1$. The initial conditions are $p_0 = 0$ and $p_1 = 1$. a) ...
0
votes
0answers
20 views

Evaluate the limit of series [on hold]

Finding the limit of series Sigma [r^1/8(n^x-1/x +r^x-1/n)]/n^x+1 as n tends to infinity , where r can take values of natural numbers and x is a constant.
1
vote
2answers
14 views

Convergence of a function involving a characteristic function of a decreasing interval

Let $f_k: \mathbb{R} \rightarrow \mathbb{R} , f_k(x) = \frac{1}{\sqrt{x}}\chi_{\left[\frac{1}{2^{k+1}},\frac{1}{2^k}\right]}(x)$ For $k \rightarrow \infty $ the interval ...
2
votes
2answers
28 views

Uniform Convergence of Series $\sum_{n=1}^{\infty}\frac{\sin(x)^n}{n}$

I'm trying to show uniform convergence of a series of complex numbers, but I'm having trouble. The series is as follows: $$\sum_{n=1}^{\infty}\frac{\sin(x)^n}{n} \rm{~~~~~~for}~~~0<x<\pi/2$$ I ...
1
vote
1answer
24 views

Proving that something diverges to infinity.

So I'm trying to prove that the sum of 1/(2k+1) diverges to infinity. I thought about doing a comparison test with the harmonic series 1/k and multiplying the harmonic series by (1/3) so it is (1/3k). ...
0
votes
0answers
32 views

Continuity on $[a,b]$ implies uniform continuity on $[a,b]$

I don't understand the step underlined in green. I understand that for any $n$ , $|f(x_n)-f(y_n)|\geq \varepsilon$ where $x_n, y_n$ satisfy the conditions given regarding a function being not ...
0
votes
0answers
19 views

Is this a Cauchy sequence?

Let $Y$ and $Z$ be Banach spaces. Define $|u|_X := |Au|_Y + |Bu|_Z$ where $A$ and $B$ are linear maps. Suppose I have a sequence $(u_n)$ such that $|Au_n|_Y \to 0$ and $|Bu_n - Bu_m|_Z \to 0$. Does ...
2
votes
1answer
32 views

Prove that if ${\{{a_n}^2}\}$ converges (${\{a_n}\}$ is monotone), thus ${\{a_n}\}$ converges and to what?

From Fitzpatrick's Advanced Calculus book: "Suppose that the sequence ${\{a_n}\}$ is monotone, i.e., either monotonically increasing or decreasing. Prove that ${\{a_n}\}$ converges if and only if ...
0
votes
2answers
21 views

Convergent sequence as series, maximum of sequence as limit

I'm currently studying for my math exams. I came across two exercises about sequences and series for which I have no clue. So any hints would be appreciated. First problem: $(a_n)_{n\in\mathbb{N}}$ ...
-1
votes
3answers
28 views

Find range of $p$ such that the series converges

let $a_n$ be a sequence of real numbers such that the series $\sum |a_n|^2$ is convergent.Find range of $p$ such that the series $\sum |a_n|^p$ is convergent. My try: To show the series it is ...
1
vote
2answers
46 views

convergence of $\sum_{n=1}^\infty\frac{1}{n} [1+\frac{1}{\sqrt{2}}+…+\frac{1}{\sqrt{n}}]$ [duplicate]

Let $t_n= \frac{1}{n} [1+\frac{1}{\sqrt{2}}+.....+\frac{1}{\sqrt{n}}]$, n=1,2... Then I am asked whether series$ \sum t_n $ converge or diverge. Also whether sequence $ t_n $ converge to zero or not. ...
3
votes
1answer
50 views

Is $a_n={\{\dfrac{1}{n^2}+\dfrac{(-1)^n}{3^n}}\}$is monotonically decreasing?

Is $a_n={\{\dfrac{1}{n^2}+\dfrac{(-1)^n}{3^n}}\}$is monotonically decreasing? In process of solving this problem, I faced to the problem of proving that $A::$: ...
0
votes
0answers
14 views

Explicit formula of a sequence [on hold]

Do you know the explicit formula for this recursively defined sequence? $(a_{n+1})=(a_n)*2+1$
8
votes
2answers
696 views

Is the product of two monotone sequences monotone?

Question: The product of monotone sequences is monotone, T or F? Uncompleted Solution: There are four cases from considering each of two monotone sequences, increasing or decreasing. CASE I: Suppose ...
3
votes
1answer
47 views

Is $\lim na_n \to 0$ sufficient for $\sum a_n$ to converge?

Is $\lim na_n \to 0$ sufficient for $\sum a_n$ to converge? Or additional criteria is required? E.g. $a_n$ needs to be positive? Is naïve comparison with $\frac {1}{n^p}$ series justifies that ? Or is ...
0
votes
1answer
16 views

Find the ratio and interval of convergence for $\sum_{n=1}^{\infty} \frac{n!x^n}{1\cdot 3\cdot 5\cdots (2n-1)}$

I believe this would diverge for $x\neq 0$. After using the ratio test I obtain (x)(n+1)(sum from 1 to n of (2n-1)/(2n+1)). Taking the limit as n goes to infinity the second term blows up and the ...
3
votes
1answer
69 views

Prove the sequence determined by $a_{n+1}={a_n\over \sin a_n}$ is convergent, and found its limit.

Let $\{a_n\}$ be a sequence defined by $0<a_1<{\pi \over 2}$, $a_{n+1}={a_n\over \sin a_n}$. $Attempt:$ $a_1>0$ and $\sin a_1>0$ and therefore the sequence begins positive and remains ...
5
votes
1answer
57 views

Evaluating $\lim_{n \to \infty} \sum\limits_{k=1}^n \frac{1}{k 2^k} $

Question: How to compute $$ \lim_{n \to \infty} \sum\limits_{k=1}^n \frac{1}{k 2^k}? $$ Here is what I have tried so far: Define $s_n=\sum\limits_{k=1}^n \frac{1}{k 2^k}$ for every index $n$, ...
-1
votes
2answers
18 views

Find closed form for a sequence using the Fibonacci and Lucas number sequences. [on hold]

Define $A_n$ as follows: $$\begin{align*} A_0 &= 6 \\ A_1 &= 5 \\ A_n &= A_{n - 1} + A_{n - 2} \; \textrm{for} \; n \geq 2. \end{align*}$$ There is a unique ordered pair $(c,d)$ such that ...
-2
votes
3answers
49 views

Easy Analysis question [on hold]

Prove $\{\sqrt{n+1}-\sqrt{n}\}$, $n ≥ 0$, is monotone, using just algebra
2
votes
3answers
29 views

Explicit (or recursive) formula of a sequence

Is there an explicit or recursive formula for this sequence starting from n=1: 1, -1 , -1 , 1 , 1 , -1 , -1 , 1 , 1 , -1 , -1 , 1 , 1 , -1 , -1 , 1 , 1 , ...
0
votes
2answers
40 views

proving a limit by definition

given a sequence $a_n=a^{\frac{1}{n}}$ for $n\in\mathbb{N}^*$, $a\in\mathbb{R},a>1$ then proof that $\lim\limits_{n\to+\infty}a_n=1$ by definition. proof: given $a_n=a^{\frac{1}{n}}$ for ...
0
votes
1answer
34 views

Convergent series proof!

Let $S$ be a non-empty subset of $\Bbb{R}$ that is bounded above. Show that there exists a sequence $(a_n)_{n\in \Bbb{N}}$ contained in $S$ (that is, $a_n \in S$ for all $n \in \Bbb{N}$) which is ...
0
votes
2answers
25 views

Determine if the series diverge or converge?

So I was wondering what is the best TEST(divergent test, alternating series,power series,ratio test or root test) that we can use for the following series :
1
vote
2answers
40 views

Infinite Sum with differential operator

How would one suggest to calculate the following sum? $\sum^{∞}_{n=1}\partial_{x}^{2n}(\frac{\pi}{2x}Erf[\frac{cx}{2}])=?$ where c is just a constant. cheers.
0
votes
0answers
12 views

need help solving this series [on hold]

i'm finding it difficult finding if this series converges or diverges. any help is appreciated. $\sum _{n=0}^{\infty }\left(3^{2+n}2^{1-3n}\right)$
0
votes
1answer
8 views

Problem with finding generating function for a sequence

Problem: Determine the generating function for the sequence $(a_k)$ given by $a_0 = 2$ and $a_k = 3a_{k-1} - 4$ for $k \geq 1$. Solution: We define $f(x) = \sum_{k=0}^{\infty} a_k x^k$ for the ...
1
vote
1answer
16 views

Prove that for every $T > \frac{\pi}{2} $, $\int_{\frac{\pi}{2}}^T \frac{cos(x)}{x}dx < 0$

I tried doing integration by parts a few times, after doing it 3 times I get the following expression: $$ \int_{\frac{\pi}{2}}^T \frac{cos(x)}{x}dx = \frac{sin(T)}{T} - \frac{1}{\frac{\pi}{2}} - ...
1
vote
1answer
25 views

Why $\frac{1}{n}\sum_{j=1}^mj^p \asymp\frac{1}{n}m^{p+1}$ as $n\to\infty$?

Why $$\frac{1}{n}\sum_{j=1}^mj^p \asymp\frac{1}{n}m^{p+1}$$ as $n\to\infty$, where $p>0$ and $a_n\asymp b_n$ if and ony if there exists a constant $c^{-1} \leq b_n/a_n \leq c$?
1
vote
0answers
45 views

Series of form $\sum_{n=1}^{\infty}f_n(x)$

I have the following series $$\sum_{n=1}^{\infty}f_n(x)$$ where $f_n(x)=\frac{(-1)^n}{n^2}\cos(n\pi x)$. I have to prove that the series converges absolutely and uniformly on the interval $[-1,1]$. ...
4
votes
2answers
42 views

$\lim_{n \rightarrow \infty} \frac {1^{a+1}+2^{a+1}+\cdots+n^{a+1}}{n.(1^{a }+2^{a }+\cdots+n^{a })} $

The value of $$\lim_{n=\infty} \dfrac {1^{a+1}+2^{a+1}+\cdots+n^{a+1}}{n.(1^{a }+2^{a }+\cdots+n^{a })} $$ Attempt: $S = \lim_{n \rightarrow \infty} \sum_{n=0} ^\infty \dfrac {k^{a+1}} {n.( 1^{a ...
5
votes
1answer
45 views

How do I solve this infinitely nested radical? [duplicate]

$\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+...}}}}$ Apparently, the answer is 3.
-3
votes
0answers
35 views

Why $\zeta(-1)=-1/12$? Doesn't defining it like this instead create problems? [duplicate]

Why $\zeta(-1)=-1/12$? As $$\zeta(-1)=\sum_{k=1}^{\infty}k\sim\infty$$How does it makes sense? Also about other $\zeta(t)\mid t<0$
1
vote
1answer
36 views

Can a divergent alternating series by rearrangement of terms be made to converge to a value?

Riemann discovered that a conditionally convergent series, through rearrangement of it's terms, can be made to converge to any value. But, if $S$ is a divergent alternating series, through ...
2
votes
1answer
29 views

How to calculate the limit of this sequence which incorporates tan?

I was revising for my pre-calculus exam, which is in two weeks time, and I started proving some sequence related theorems. I got interested in limits and I started deepening the concept. I got to a ...
0
votes
0answers
24 views

Is it possible to show the uniqueness of formula for solution?

The motivation to this question can be found in: Show that any sequence $(u_{n})$ must tends to infinity as $n→∞$ My question is: Is it possible to show the uniqueness of the formula for the ...
2
votes
0answers
32 views

Problem with constants of $\sin(x)$ and using partial sums (Basel Problem)

I'm working on the Basel problem, and in my working I have $\sin(x)=Ax(x-\pi)(x+\pi)(x-2\pi)(x+2\pi)...$ But I think I have a problem with the coefficient of x - wouldn't it be infinitesimal, if we ...
1
vote
1answer
40 views

Convergence of $\sum_{n=1}^\infty n^s(\sqrt {n+1} - 2 \sqrt n + \sqrt {n-1})$

Convergence of $$\sum_{n=1}^\infty n^s(\sqrt {n+1} - 2 \sqrt n + \sqrt {n-1})$$ Attempt: $\sum_{n=1}^\infty n^s(\sqrt {n+1} - 2 \sqrt n + \sqrt {n-1}) \sim \sum_{n=1}^\infty n^s( \sqrt n )$ As ...
3
votes
1answer
71 views

Prove that if $a_n>0$ and $\sum a_n$ converges then $\sum (\frac {b_n}{a_n})$ converges

Let $\{a_n\},\{b_n\}$ be two sequences such that for each $n$ we have $e^{ a_n }= a_n + e^{b_n }$. Show that if $a_n>0$ and $\sum a_n$ converges $\implies \sum \left(\frac {b_n}{a_n}\right)$ ...
0
votes
1answer
37 views

Limit of sequence $ S_n $=$\int_0^1 \frac{nx^{n-1}}{1+x}dx $ [duplicate]

Let $ S_n $=$\int_0^1 \frac{nx^{n-1}}{1+x}dx $ for n$ \ge $1 then as n tends to infinity sequence tends to: 1.0 2.1 3. 1/2 4. Infinity Is there any other way, than to first do integration, and then ...
0
votes
0answers
32 views

Is there a way to reverse the ratio test?

My question arises from the following problem: Let $ a_n $ be a real series, so that $ \sum_{n=1}^ \infty a_n $ converges and $a_n \ge 0 $ and $a_n$ monotonously decreasing. It is to prove: $ ...
2
votes
0answers
32 views

$\lim_{n \to \infty}\prod_{k=1}^n \cos(k\sqrt{\frac{3}{n^3}} t) = e^{- \frac{t^2}{2}}$ [duplicate]

Is it true that $$\lim_{n \to \infty}\prod_{k=1}^n \cos\left(k\sqrt{\frac{3}{n^3}} t\right) = e^{- \frac{t^2}{2}}$$ ? How to proceed ?
0
votes
1answer
22 views

Closed form solution of a summation

First off I have absolutely no clue what I'm doing, my notes given for this course do not explain anything and I'm not sure if I'm doing this properly so I'm looking for help and an explanation on how ...
0
votes
0answers
11 views

Convergence of a Sequence in l^1 space

If a,b are complex numbers, and $k\ \epsilon$ N, the sequence $x_k = a+b^k$ will belong to $l^1(N) = \lbrace (c_0,c_1... ) : \sum_{k=0}^{\infty} |c_k| < \infty \rbrace$ for which a,b? a and b ...