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

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-1
votes
2answers
32 views

Convergence of $\sum_{n=1}^{\infty}\frac{(n!)^n}{(n^n)^2}$

Test the convergence of the following series $$\sum_{n=1}^{\infty}\frac{(n!)^n}{(n^n)^2}$$
9
votes
0answers
32 views

Rearrangements that never change the value of a sum

Which bijections $f:\{1,2,3,\ldots\}\to\{1,2,3,\ldots\}$ have the property that for every sequence $\{a_n\}_{n=1}^\infty$, $$ \lim_{n\to\infty} \sum_{k=1}^n a_n = \lim_{n\to\infty} \sum_{k=1}^n ...
4
votes
2answers
86 views

Weird thing about $\sum_{k=2}^{\infty}(-1)^k\zeta{(k)}$

Consider the sum $S=\sum_{k=2}^{\infty}(-1)^k\zeta{(k)}$. By a simple manipulation, we can show: $$ ...
0
votes
0answers
33 views

Need help solving min, max, inf and sup of sequence!

We've been given the sequence $x_n=(-1)^n \cdot \frac{\sqrt{n}}{n+1}+\sin \frac{n \pi}{2}$. I have to find the min, max, inf and sup (if they exist), and also find the points of accumulation. Any ...
6
votes
1answer
55 views

Closed form of series involving Fibonacci numbers

Let $F_n$ denote the $n$-th Fibonacci number and $\phi$ be the golden ratio, that $\phi = \frac{1+\sqrt{5}}{2}$. Find a closed form for the sum: $$\sum_{n=0}^{\infty} \frac{1}{(5\phi)^n(n+2)} ...
1
vote
3answers
42 views

What does it mean for a function to “preserve the limits of sequences”?

I've translated the English Wikipedia page "Limit of a sequence". What does the following statement mean? In fact, any real-valued function f is continuous if and only if it preserves the limits ...
2
votes
1answer
84 views

Convergence of $\sum \frac{2n+1}{(n^2+n)^n}$

I have to choose the right option: The series $$\sum_{n\geq 1} \frac{2n+1}{(n^2+n)^n}$$ a. Converges to 1. b. Converges to a number >1. Using ...
1
vote
2answers
53 views

Trouble with finding the limit of this sequence

Well I was trying to find the limit of - $$ \lim_{x\rightarrow \infty } \lim_{n\rightarrow \infty} \sum_{r=1}^{n} \frac{\left [ r^2(\sin x)^x \right ]}{n^3}$$ obviously $$ ...
-1
votes
0answers
24 views

infinite limit of factorial funtion [on hold]

What will be the limiting value of $$\lim_{n \to \infty} \, \sum_{j=0}^{\left[\frac{n}{c}\right]}\frac{(1+a)_{bn}}{(n-cj)!}$$ where $a,b,c \in \mathbb{N}$ and $(x)_{m}$ is the Pochhammer symbol ?
10
votes
2answers
120 views

$\lim x_n^{x_n}=4$ prove that $\lim x_n=2$ [duplicate]

Let $(x_n)$ be a sequence of real numbers, such that: $\lim x_n^{x_n}=4$, prove that $\lim x_n=2$ I'm not sure if my proof is right. I assumed that $\lim x_n $ isn't 2 and using Cauchy's criterion: ...
-6
votes
0answers
45 views

Julia and Mandelbrot Sets [on hold]

I need to know how escape, prisoner, Julia and Mandelbrot sets work. Are they all in one sequence or are they separate.
3
votes
1answer
50 views

Prove or disprove that a series is convergent

I was given the following task which I struggle with. Prove the following statement, or disprove it by giving a counter example: if $\sum_{n=1}^\infty a_n$ is convergent then $\sum_{n=1}^\infty ...
2
votes
2answers
79 views

Convergence of Sequence

If I know that the sequence $\{a_n\}$ converges to $a$, then to prove that the sequence $\{ca_n\}$ (for a constant $c$) converges to $ca$, I would basically want $|ca_n - ca| < \epsilon$... So, to ...
1
vote
0answers
33 views

Can someone suggest some reference for a particular kind of infinite series

Can someone suggest some reference for the properties of the series $\lim_{n \to \infty} \sum_{i = 0}^{n} f(n, i)$ For example, when can we apply approximations to $f(n, i)$ that only works when $n ...
1
vote
1answer
20 views

A question about fixed-point iteration sequence of a two times continuously differentiable function

I am stuck at this problem: Let $g:[a,b]\to[a,b]$ be a 2 times continuously differentiable function that satisfy: for all $x\in [a,b]$, $g''(x)\neq 0$ And let $s$ be an arbitrary fixed-point of ...
2
votes
3answers
67 views

Why would the cubic have $5$ roots?

The polynomial $P(x)$ is cubic. What is the largest value of $k$ for which the polynomials $Q_{1}(x) = x^{2}+(k-29)x-k$ and $Q_{2}(x) = 2x^{2}+(2k-43)x+k$ are both factors of $P(x)$? $P(x) = ...
6
votes
1answer
98 views

Find the sum of the series below

Find the sum $$(1\cdot2)+(1\cdot3)+(1\cdot4)+\cdots+(1\cdot2015)+(2\cdot3)+(2\cdot4)+\cdots+(2\cdot2015)+\cdots+(2014\cdot2015)$$ What I have tried... We are looking for ...
1
vote
1answer
72 views

Convergence to $0$ of a certain series.

I was wondering whether or not the following holds - I didn't manage to get anywhere using standard tricks from elementary analysis. ...
2
votes
2answers
72 views

On the sequence of function $f_n(x)=n^2x(1-x^2)^n $

Is the sequence of functions $f_n(x)=n^2x(1-x^2)^n $ uniformly convergent over $[0,1]$ ? Is the series $\sum_{n=1}^{\infty} f_n(x)$ convergent with $0<x<1$ ? Is the series uniformly convergent ...
1
vote
1answer
47 views

Condition on p for convergence of $\sum{\frac{1}{n(\log(n))^p}}$

For what values of $p$ is the series $\sum{\frac{1}{n(\log(n))^p}}$ divergent and for what values it is convergent?
5
votes
1answer
158 views

Can we find this limit?

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence such that $a_1=1$ and $a_{n+1}=\sqrt{a_n^2+a_n}$ for $n\geq 0$. Is it possible to find $\displaystyle\lim_{n\to\infty}\frac{a_n}{n}$ ? I have no any idea. ...
1
vote
3answers
76 views

Series convergence $x+\frac{2^2x^2}{2!}+\frac{3^3x^3}{3!}+\frac{4^4x^4}{4!}+\cdots$ [on hold]

Choose the right option. The series $x+\dfrac{2^2x^2}{2!}+\dfrac{3^3x^3}{3!}+\dfrac{4^4x^4}{4!}+\cdots$ is convergent if a. $0<x<1/e$ b. $x>1/e$ c. $2/e<x<3/e$ d. $3/e<x<4/e$ ...
0
votes
0answers
47 views

Is it possible to find the exact value of this

Is it possible to find the exact value of the infinite series ? $$\sum_{n=1}^\infty \frac{2^n}{(1+\sqrt{2})^n+1}$$ I have no any idea. Thank you for helping.
1
vote
0answers
13 views

Determining an integer to approximate error using the Alternating Series Test

I need to determine the smallest value m, such that: $$ \left |\int_0^{0.1} arctan(x^2) dx - \sum_{n=0}^m (-1)^n \frac{(0.1)^{(4n+3)}}{(4n+3)(2n+1)} \right| < 10^{-8} $$ Using the Alternate ...
0
votes
0answers
30 views

Proof that the sum of a certain infinite series can be bounded to zero

$\forall 0 < \alpha < 1$, there exists $\lambda > 0$, $k > 0$, s.t. $$ \lim_{n \to \infty} \sum_{w = 1}^{\lambda n} \binom{n}{w} \frac{1}{2^{\alpha n}}\left(1 +\left(1 - ...
0
votes
1answer
19 views

Summation of finite power seires

Is it possible to find a close form solution for $S_1$. $S_1$ is defined as follows: $S_1=\sum_{k=b}^{\infty}\frac{x^k}{k!}$ ; Where $0<x<b<\infty$ If $b=0$ then $S_2 = e^x$. But how do we ...
-2
votes
0answers
17 views

Cauchy second theorem [on hold]

Where to use Cauchy second theorem I mean in it can be used only in the case of powers and factorials or somewhere else also.
1
vote
1answer
37 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
51 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
80 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
votes
2answers
80 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
votes
3answers
52 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
110 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
180 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
votes
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
46 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
4answers
73 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
vote
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
36 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. ...
-2
votes
0answers
74 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 ...