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

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1
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1answer
38 views

Tough problem on sum of infinite series [on hold]

I've been working on the problem for quite a while but have no idea how to approach it. This proposition arises from a practical probabilistic bound problem, but it seems very deep. Lots of thanks to ...
2
votes
4answers
66 views

Does $\displaystyle\sum^{\infty}_{n=1}\left(\frac{n!}{n^n}\right)$ converge or diverge? [duplicate]

Does $\displaystyle\sum^{\infty}_{n=1}\left(\frac{n!}{n^n}\right)$ converge or diverge? I've tried the ratio test, but i'm unsure if I can continue this way. ...
0
votes
1answer
55 views

Is that form of Cesàro's theorem correct?

First, for sequences, we know that : "If a sequence $(a_n)_{n \ge 1}$ converges to $l\in \mathbb{R}$, then the sequence $(b_n)_{n \ge 1}$ defined by : $b_n=\frac{1}{n} \sum \limits_{k=1}^{n}a_k$ ...
3
votes
0answers
87 views

Is there a closed-form approximation to a band-limited sawtooth?

A partial Fourier Series with no coefficients is equal to the closed form expression: $${A \over n} \sum_{k=1}^n \cos(k\theta) = {A \over 2n} \left\{{\sin([2n + 1]\theta/2) \over \sin(\theta/2)} - ...
2
votes
4answers
61 views

Superior limit of a certain sequence

Let $(x_n)$ be the sequence: $$\{1, 2, 1 + \frac12, 2 + \frac12, 1 + \frac13, 2 + \frac13, \ldots \}$$ I (think I) understand why $\lim \inf x_n = 1$, but I'm not sure why $\lim \sup x_n =2$. Any ...
1
vote
2answers
24 views

Can we replace the limit of a sequence with that of a function?

Let $f$ be a function defined in $[1,\infty]$. If $\lim_{x\to\infty}f(x) = L$ and $a_n = f(n)$ for integer $n\ge 1$ then $\lim_{n\to\infty}a_n = L$. Found this theorem in many references, but ...
2
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2answers
84 views

Find the limit of an infinite series

My intuition was to try and see if the series is a Riemann Sum of a function and then see what happens but I can't really see which function fits here. Thanks!
0
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3answers
53 views

What is the limit of this sequence as n->infinity? [on hold]

Find the limit of the following sequence $n^{\ln(n)/n}$ as $n\to\infty$? Please answer without using L'Hopital
0
votes
5answers
112 views

Find limit of the following sequence?

Find the limit of $\frac{\log(n+1)}{\log(n)}$ where $n\rightarrow\infty$. Here $n$ is a natural number so I guess we can't use L'Hopital
9
votes
2answers
181 views

Proving a sequence converges when combinations of consecutive terms converge

Problem: Let $\{x_n\}$ be a sequence of real numbers such that $$\lim_{n\to\infty} 2x_{n+1}-x_n=L \in \mathbf{R}.$$ Prove that $x_n \to L$ as $n\to\infty$. I can see that if $\{x_n\}$ converges to a ...
2
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3answers
39 views

calculate two-fold difference

These are a series of numbers that increase two folds: $$0.125, 0.25, 0.5, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024$$ If I pick up two numbers, say $0.5$ and $128$, I want to know know how may ...
2
votes
1answer
47 views

Geometric series and complex numbers [duplicate]

I'm new to this site, english is not my mother tongue, and I'm just learning LaTeX. I'm basically a noob, so please be indulgent if I break any rule or habits. I'm stuck at proving the following ...
7
votes
1answer
127 views

Proving convergence for a series.

Let there be given that $\displaystyle\sum_{n=1}^{\infty}a_{n}$ is a convergent series and $a_{i}$ is not negative for every $i\in\mathbb{N}$. And i want to prove that this series is also convergent ...
-1
votes
3answers
77 views

The sum of the series $\sum_{n=1}^{\infty}\sin^n(k)$

What is $$\sum\limits_{n=1}^{\infty}\sin^n(k)?$$ Can you find what is the sum of that series. It is convergent not divergent. What if $k=\frac{\pi}{6}$?
0
votes
1answer
39 views

What to know before solving sequence and series problems?

Talking about How to solve the sequence: $87, 89, 95, 107, ?, 157$ for example, I read the hint: The difference between each term goes like this: 2,6,12. Can you notice any pattern? Based on it, ...
0
votes
4answers
55 views

Find the Limit of the sequence $\frac{(2^n)(n!)}{(2n+1)!}$

Find the Limit of the sequence $\frac{(2^n)(n!)}{(2n+1)!}$ I'm not sure how to approach this problem. I tried the squeeze method, but could not figure it out.
3
votes
1answer
65 views

How to resolve the issue of two sequences converging to zero for $n, m \to \infty$?

My question is motivated by the following exercise in probability theory: Let $X_n \to X$ in probability and $X_n \geq Y$ a.s. Show that $X \geq Y$ a.s. I noticed that for all $n, m \in ...
1
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0answers
21 views

Tighter upper bounds with ratios of powers of norms

This question arises in concentration or sparsity measures for finite sequences. Given $x\in \mathbb{R}^K$ and $1 \le r < s$, i try to find a tight upper bound for $$\psi_{r,s}(x) = \frac{\sum_1^K ...
0
votes
1answer
34 views

Finding the series for $(\ln(1+z))^2$

So, I'm supposed to use product of infinite series methods to find the series for $(\ln(1+z))^2$. I'm given that the answer has the form $$z^2 \sum_{l=0}^{\infty} c_l z^l$$ and I'm given that the ...
0
votes
1answer
43 views

If $\displaystyle\lim_{n\to\infty}|x_n|^{1/n} = L < 1$ then $\displaystyle\lim_{n\to\infty}|x_n|=0$. [on hold]

Let $(x_n)$ be a sequence of real numbers such that $\displaystyle\lim_{n\to\infty}|x_n|^{1/n} = L < 1$. Show that $\displaystyle\lim_{n\to\infty}|x_n|=0$. Remark: Not use that exist $0<r<1$ ...
3
votes
1answer
37 views

Convergence of fixed-point iteration for $p$ times continuously differentiable function

I am stuck at this problem: Let $\alpha\in\Bbb{R}$ be some number that satisfies $g(\alpha)=\alpha$ for some function $g$ that is $p$ times continuously differentiable on some neighborhood of ...
2
votes
2answers
52 views

Terms of a certain recurrence

Let $a_1, a_2\dots $ be a sequence of reals such that $a_1 = a_2 = 1$, and $$a_{n + 2} = \frac{a_{n + 1}^3 + 1}{a_n}$$ for $n \ge 1$. It appears to be the case that all of these values are integers. ...
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votes
1answer
100 views
+100

What are some tricks to solve Progressions quickly?

Recently while solving few questions related to Progressions(specifically, A.P.), I realized one thing that in question like, "Find the sum of following series" and suppose the terms are up to ...
-1
votes
0answers
50 views

Can someone answer and explain the missing term of this sequence problem? [on hold]

$8,28,?,14,14,6,17,3,...$ Which one is the answer and why? a) $9$ b) $11$ c) $10$ d) $12$
5
votes
3answers
73 views

Showing that the sequence $x_n=\frac{1}{3}x_{n-1}(4+x_{n-1}^3)$ where $x_0=-0.5$ quadratically converges

I am stuck at a point in solving this problem: Show that the sequence defined by: For all $$n\in\mathbb{N}, x_n = \begin{cases} -\frac{1}{2}, & \text{if $n=0$} \\ ...
1
vote
3answers
70 views

find the complex number $z^4$

Let $z = a + bi$ be the complex number with $|z| = 5$ and $b > 0$ such that the distance between $(1 + 2i)z^3$ and $z^5$ is maximized, and let $z^4 = c + di$. Find $c+d$. I got that the ...
3
votes
2answers
87 views

pointwise almost everywhere convergent subsequence of $\{\sin (nx)\}$

Can you prove or disprove that the sequence $\{\sin (nx)\}$ has a pointwise convergent almost everywhere subsequence with respect to the Lebesgue measure on $\mathbb{R}$ ? Edit: I am adding my ...
-6
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1answer
49 views

What is the next value in this sequence? [on hold]

16 104 572 2574 9009 22522.5 ?
6
votes
1answer
104 views

FRACTRAN for natural numbers

Is there a simple analogue of FRACTRAN that maps a natural number to a natural number, instead of mapping a list of fractions to a natural number? One could use Gödel encoding to translate FRACTRAN ...
1
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1answer
36 views

What is the definition of the absolute convergence of an infinite product $(1+a_1)(1+a_2)(1+a_3)\cdots$?

For an infinite product $(1+a_1)(1+a_2)(1+a_3)\cdots$, whats the definition of convergence and absolute convergence? Why the absolute convergence corresponding to the absolute convergence of sum ...
-1
votes
2answers
33 views

What is the difference of the greatest of the limits $\overline{\lim}_{n\to \infty}$ and the least of the limits $\underline{\lim}_{n\to \infty}$?? [closed]

What are exactly these three limits for an infinite series $x_n$? $$\overline{\lim_{n\to \infty}} x_n$$ $$\underline{\lim}_{n\to \infty} x_n$$ $$\lim_{n\to \infty} x_n$$ Can they be different from ...
0
votes
1answer
56 views

Calculat sums of the form $\lim_{n\rightarrow\infty}\sum_{i=0}^{n}\left(\frac{i}{n}\right)^{f(n)}$

Problem: calculate the sums of the form: $$\lim_{n\rightarrow\infty}\sum_{i=0}^{n}\left(\dfrac{i}{n}\right)^{f(n)}$$ Inspiration: one problem lets us prove that ...
0
votes
2answers
34 views

Trying to understand why 2 times the sum of consecutive integers from 0 to n is equal to n times n+1

I am sorry if this question ends up being a duplicate, as I am having a bit of a challenge explaining it to myself well enough to know how to query it. There is a Facebook meme that has been ...
1
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0answers
22 views

Intriguing Poisson sum with hyperbolic function

I've been playing with lots of Poisson sums lately, and I thought this one to be interesting:\ $$\sum_{k\in\mathbb{Z}}\left(\frac{1}{(k+x)\sinh{(k+x)\pi q}}-\frac{1}{\pi q (k+x)^2}\right)$$I want to ...
1
vote
6answers
75 views

For which values of $x$ does this series converge?

For which values of $x$ does the series presented below converge? $$\sum_{n=1}^{+\infty}\frac{x^n(1-x^n)}{n}$$ Neither the root test nor the ratio test is of much help - I've tried for ...
2
votes
1answer
43 views

Finding a general term for the sequence

Find a general term in simplest form for the sequence: $$2, 1, -4, 7, -10, 13, -16$$ This is what I tried: $a_n = a_1 + (n - 1)\cdot d$ $a_n = 2 + (n-1)\cdot 3$ $a_n = (-1)^n (2+(n-1)\cdot 3)$ which ...
0
votes
0answers
12 views

Double Sum of Series Unchanged When Each Term Scaled

Can the following ever hold: $$\sum_{i}\sum_{j}(-1)^ja_{i,j}=\sum_{i}i(b_{i})\sum_{j}(-1)^ja_{i,j}$$ where $b_{i}>1/i$ for all $i$? What if you tack on the fact that ...
3
votes
1answer
41 views

Sum involving the number of zeros of $k$

I'm currently interested in sums involving digit-functions. Especially, I'd like to calculate the following sum: $$ s=\sum_{k=1}^{\infty}{\frac{a_0(k)}{k\left(k+1\right)}} $$ Where $a_0(k)$ is the ...
0
votes
2answers
25 views

Series And Sequences Question

Can someone help show me what I did wrong? The question is "Find the sum of the first ten terms in this geometric series: $-5, 10, -20, \ldots$ I plugged it into this equation: $S_n = a(r^n ...
0
votes
2answers
67 views

Help needed with the integral of an infinite series

Can you please help me with the integral of this series? I came across it in a signal processing paper and haven't been able to figure out the solution myself. $$ ...
1
vote
1answer
46 views

Help needed with the integral of an infinite series

Can you please help me with the integral of this series? I came across it in a signal processing paper and haven't been able to figure out the solution myself. $$ ...
1
vote
0answers
36 views

Proving an inequality involving discrete variables

I'm trying to show that the following inequality holds $$ \frac{1-x^{n}}{1-x^{n+1}}\geq\frac{\sum_{i=0}^{n-2}x^{i}(1-x_{1}^{n-(i+1)})}{\sum_{i=0}^{n-1}x^{i}(1-x_{1}^{n-i})}, $$ where $n$ is a ...
2
votes
1answer
54 views

Notion of fixed point for the sequence $x_{n+1}=f_n(x_n)$

Let $(X,d)$ be a complete metric space and let $f_n:X\to X,\ n\in \mathbb{N}$ be a sequence of contractions. Is there any notion of fixed point for the following sequence? $$x_{n+1}=f_n(x_n),\ ...
1
vote
1answer
50 views

Rearranging a series' terms

So i am asked to rearranje the terms in this series: $$ \sum_{i=1}^\infty \frac{(-1)^{n+1}}{n} = 1- \frac 12 +\frac13-\frac14+... $$ so that the sum of the series is equal to 0. I've seen the ...
2
votes
1answer
47 views

Find a constant $C$ such that $ \Bigg| \frac{\prod_{i=0}^{k-1} (n-i) }{n^k} - 1 \Bigg| \leq \frac{C}{n}, \forall k \leq n$

Consider the following: $$ \Bigg| \dfrac{\prod_{i=0}^{k-1} (n-i) }{n^k} - 1 \Bigg| \leq \frac{C}{n}, \forall k \leq n $$ How to find an expression for $C$ independent of $k$ and thus $n$? It arises ...
2
votes
2answers
51 views

How to show that the following series converges to 1

Let $f$ be a function on $\mathbb{R}$, non-zero only on $[0,2)$. In particular $f(x)=1,x\in[0,1]$ and decreasing to zero, starting from $x=1$. Let $g(x)=f(x)-f(2x)$. Show that $$\sum_{j=0}^\infty ...
0
votes
2answers
70 views

The Laurent Series of $\dfrac{e^z}{z^2-1}$

The Laurent Series of $\dfrac{e^z}{z^2-1}$ At $z=1$ As we seek for powers of $z-1$, note that: $$e^z=e\cdot e^{z-1}=e(1+(z-1)+\dfrac{(z-1)^2}{2!}+\dfrac{(z-1)^3}{3!}+...)$$ So: ...
3
votes
1answer
46 views

Does this sequence of functions converge uniformly?

So the questions says, let $a_n$ be a sequences of real numbers such that $\limsup |a_n| = 0$. Let $X = [0, 1]$ and for each $n \in \mathbb{N}$ the function $\space$ $f_n :$ $X \mapsto ...
0
votes
0answers
14 views

What are the coefficients of Modified Fourier Bessel series?

What are the co-efficients of Modified Fourier Bessel series? $$\phi g(r,z,Φ) = \sum_{v=1}^∞ \sum_{m=1}^∞ \sin\left(\frac{mπz}{L}\right) I_v \left(\frac{mπr}{L}\right)\left(A_{vm} \cos(vΦ) + B_{vm} ...
3
votes
1answer
63 views

Proving that a sequence converges or diverges

Prove or disprove that there is a sequence $n_k$ of positive integers (that is not constant) such that both $\cos(2n_k!)$ and $\sin(2n_k!)$ converge. I think that the series diverges but I am not ...