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

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3
votes
2answers
155 views

Upper bound for an infinite series with a square root

I've been trying to find a tight upper bound for the series $$S (x) = e^{-x} \sum_{k=0}^{\infty} \frac{x^k}{k!} \sqrt{k+1}$$ So far, I've managed to get a reasonable bound for small values of $x$ by ...
0
votes
1answer
84 views

Finding a generating function for a pattern

I was working on this projecteuler.com problem, and I was very interested by the premise. Essentially, given n terms, find an ...
1
vote
2answers
123 views

On defining sequences

Can two infinite sequences be "concatenated"? Two examples: \begin{align*} S &= (2,4,6,8,10\ldots,1,3,5,7,9\ldots)\\ \\ T &= (2,3,7,13,19,\ldots,5,11,17,23\ldots) \end{align*} My hunch is ...
1
vote
2answers
453 views

When can one conclude that a sequence of uniformly bounded equicontinuous functions converges uniformly?

Let $~f_{n}: [0,1] \rightarrow \mathbb{R}$ be a sequence of smooth functions that are uniformly bounded and equicontinuous. By Arzela Ascoli theorem we know that a subsequence $\{ f_{n_k} \} $ ...
1
vote
2answers
496 views

What is $\sup(\sin(n))$? [duplicate]

My classmate asked a question during lecture about our discussion of bounded sequences, particularly the sequence $\sin(n)$. His question was, What is $\sup(\sin(n))$?
8
votes
3answers
196 views

Proving that the limit of a sequence is $> 0$

Let $u$ be the complex sequence defined as follows : $u_0=i$ and $ \forall n \in \mathbb N, u_{n+1}=u_n + \frac {n+1-u_n}{|n+1-u_n|} $ . Consider $w_n$ defined by $\forall n \in \mathbb ...
2
votes
0answers
80 views

Upper Bound on $\frac{1}{1-\beta u}-\sum\limits_{n=0}^{\infty}\frac{e^{-u}u^n}{n!(1-\beta(n+1))}$

Is there any procedure to find an upper bound of the following expression? $$\frac{1}{1-\beta u}-\sum_{n=0}^{\infty}\frac{e^{-u}u^n}{n!(1-\beta(n+1))}$$ Here $u,\beta\in\mathbb{R},\ u>1,\ ...
1
vote
2answers
128 views

Sequence limit!!

Hi I have a question about a proof of limit. In my text book, there is an example to prove $\lim \frac{n^2+3}{n+1}=+\infty$. The definition of a diverge sequence is as follows. $$\forall M>0, ...
6
votes
2answers
340 views

Prove that the sequence given by $c_n = \sqrt{1+c_{n-1}}$ converges and find the limit

Let $c_1 = 2$, and for $n > 1$, let $c_n = \sqrt{1+c_{n-1}}$. Prove: (by induction) that $c_n < 2$, for $n > 1$. (by induction) that {$c_n$} is monotonically decreasing. that ...
1
vote
0answers
29 views

Solving a Series [duplicate]

1, 5, 13, 27, 48, 78, 118, 170, 235, 315, 411. Background: This comes from a question of how many total triangles are there in a large triangle whose base equals n of the smallest possible ...
0
votes
2answers
168 views

Question on geometrical proof of Geometric Series

The following image is from Geometric Series Proofs: An Annotated Bibliography. Please explain why it is said that: "$ON$ is the limit of the sum $1+x+\dots$." Thank you. Edit: I guess what ...
0
votes
2answers
987 views

Prove the convergence of the geometric series using $\epsilon$, N definition

Show $| \sum_{n=0}^{\infty}x^n - \frac{1}{1-x} | < \epsilon$ using the definition of convergence when |x|< 1.
0
votes
2answers
101 views

Prove $\left(\frac{1}{n}+\frac{(-1)^n}{n^2}\right)$ converges to $0$ as $n\to\infty$

Using the formal definition of convergence of a sequence, show that the sequence converges to 0 as n tends to infinity. So we want to show that for every $\epsilon>0$, there exists $N$ such that ...
5
votes
1answer
408 views

Convergence of sequences of inverse functions

Let $(X, \phi)$ and $(Y, \sigma)$ be metric spaces, and let $f, f_1, f_2, \ldots$ bijective function with inverse functions $g, g_1, g_2, \ldots$ $f_n \to f$ pointwise for $n \to \infty$. And all ...
0
votes
1answer
127 views

Showing Non-Uniform Convergence of a Series

I've been asked to show $\displaystyle\sum_{k=1}^{\infty}\frac{x^2}{k^2+x^2}$ $\forall x \in \mathbb{R}$ does not uniformly converge in $\mathbb{R}$. I do know and have already shown it plainly ...
7
votes
0answers
66 views

Infinite series with prime number [duplicate]

I know the $\sum_{n=1}^\infty \frac{1}{\text{Prime[$n$]}}$ does not converge, but what about the following series? $$\sum_{n=1}^\infty\frac{1}{\text{Prime[Prime[$n$]]}}$$ (Where $\text{Prime[$n$]}$ ...
6
votes
3answers
500 views

How do I write this sum in summation notation?

$$\sum_{n=?}^\infty \left(\frac{x^n}{?}\right) = \frac{x^0}{1} + \frac{x^1}{x^2 -1}+\frac{x^2}{x^4 - x^2 +1}+\frac{x^3}{x^6 -x^4 + x^2 -1}+\frac{x^4}{x^8-x^6 +x^4 - x^2 +1}+\cdots$$ I am pretty sure ...
2
votes
1answer
125 views

Is a sequence of $L^p$ with a weakly convergent subsequence weakly convergent?

Consider $u_n$ a bounded sequence in $L^{p}(\Omega)$ where $\Omega \subset R^n$ open and bounded. Suppose that exists a subsequence $u_{n_j}$ that converges weakly to a function $u $ in ...
2
votes
1answer
135 views

Completely monotone condition

I've stucked in this problem for a while. So I hope someones could give me suggestions. Consider the function: $$ f( r ) = \frac{e^{- br}}{1 + ce^{ - dr}} = \sum\limits_{n = 0}^\infty {( - c)^n}e^{ - ...
1
vote
1answer
73 views

limit of a recursive sequence:2

Let $$x_k = \frac{A}{1-C} x_{k-1} + \frac{B}{1-C}x_{k-2},$$ where $A, B, C$ are positive reals such that $A + B + C =1$. Let $$x_1 = 1$$ and $$x_2 = 1 + y,$$ with $y$ is positive. Which conditions ...
21
votes
2answers
1k views

Proving that $\frac{\pi}{4}$$=1-\frac{\eta(1)}{2}+\frac{\eta(2)}{4}-\frac{\eta(3)}{8}+\cdots$

After some calculations with WolframAlfa, it seems that $$ \frac{\pi}{4}=1+\sum_{k=1}^{\infty}(-1)^{k}\frac{\eta(k)}{2^{k}} $$ Where $$ \eta(n)=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k^{n}} $$ is the ...
1
vote
1answer
25 views

On transcendentals from sequences

Say we have a sequence of integers $N_{k}$ where $k \in \Bbb N$. Can $\limsup_k \sqrt[k]{N_{k}}$ be transcendental?
3
votes
1answer
152 views

Monotonicity of the sequence $ ( F_n^{\frac{1}{n}} ) $, where $ ( F_n ) $ is the Fibonacci sequence

Let $ F_n = F_{n-1} + F_{n-2} $ with $ F_0 = 1 $, $ F_1 = 1 $ (the Fibonacci sequence). I would like to know whether $ F_n^{\frac{1}{n}} $ is monotonically increasing in $ n $. It is not difficult to ...
0
votes
2answers
202 views

An exception to Taylor Series

According to Taylor Series, $$f(x) = \sum_{n=0}^\infty \dfrac{f^{(n)}(a)}{n!}*(x-a)^n $$ However, $\dfrac{1}{x}$,$\dfrac{1}{x^2}$, etc. are not applicable. I tried to do the following: $$ ...
5
votes
1answer
261 views

Closed form of $\sum_{n=1}^\infty \frac{2^{3n}}{2^{4n}+2^{2n}} $?

$$\sum_{n=1}^\infty \frac{2^{3n}}{2^{4n}+2^{2n}} $$ I know it converges thanks to WolframAlpha, but want to know exactly what it converges to. I can calculate the decimal, but don't know how to begin ...
1
vote
2answers
62 views

Convergent/divergent series

I need to prove that series are divergent/convergent: $\displaystyle\sum_{n=2}^{\infty}(n\sqrt{n}-\sqrt{n^3-1})$ I tried using Limit comparison (with $1/n$), Root and Ratio tests, but they gave no ...
10
votes
1answer
256 views

How prove $a_{n}=1!+2!+\cdots+n!$ contains infinitely many prime factors

show that $$(n+1)b_{n+1}-b_{n}=n+1,b_{1}=1,a_{n}=n!b_{n}$$,then the sequence $\{a_{n}\}$ contains infinitely many prime factors my idea:since $$(n+1)b_{n+1}-b_{n}=n+1$$ $$\Longrightarrow ...
0
votes
3answers
67 views

$\sum \frac{a_k}{p^k}$ converges to a real number in $[0,1]$

Given $p \geq 2$, $$ \sum_{k=1}^{\infty} {\frac{a_k}{p^k}} , a_k \in \{0,1,...,p-1\}$$ converges to a real number in $[0,1]$ I was thinking maybe using geometric series the above series can written ...
6
votes
3answers
258 views

Some formulae for a periodic sequence $-1,-1,1,1…$?

Some formulae for a periodic sequence? when $T = 2$, we have $-1,1,-1,1,-1,1,\text{...}$, the formula is $$\begin{align*}(-1)^n\end{align*}$$ when $T = 4$, we have ...
2
votes
2answers
112 views

Calculate limit of a series

Calculate $$ \lim_{n \to \infty} \sum \limits_{k=1}^n \frac{n}{k^2 - 4n^2} $$ or prove it doesn't exist.
2
votes
1answer
84 views

How to evaluate the infinite series: $ \dfrac 1 {3\cdot6} + \dfrac 1 {3\cdot6\cdot9} +\dfrac 1 {3\cdot6\cdot9\cdot12}+\ldots$

The infinite series is given by: $$ \dfrac 1 {3\cdot6} + \dfrac 1 {3\cdot6\cdot9} +\dfrac 1 {3\cdot6\cdot9\cdot12}+\ldots$$ What I thought of doing was to split the general term as: $$\begin{align} ...
4
votes
1answer
124 views

How to prove that $_2F_1 \left(\frac{n+2}{2},\frac{n+1}{2};\frac{3}{2};-\tan^2 z\right) = \frac{\sin nz \cos^{n+1}z}{n\sin z}$?

Formula 9.121.19 of I. S. Gradshteyn and I. M. Ryzhik. - Table of Integrals, Series, and Products states that $$_2F_1 \left(\frac{n+2}{2},\frac{n+1}{2};\frac{3}{2};-\tan^2 z\right) = \frac{\sin nz ...
1
vote
0answers
336 views

Proof of Toeplitz's theorem.

Let $a_n$ be a real sequence convergent to $a \in \mathbb{R}$. Let $t_{k,n}$ (where $1\le k \le n$) be a sequence of weights such that: $$(I) \quad \forall k \lim_{n \to \infty}t_{k,n} = 0$$ $$(II) ...
2
votes
3answers
102 views

$x_1=0,\,x_{2n}=\frac{x_{2n-1}}{2},\,x_{2n+1}=x_{2n}+\frac{1}{2}$ Find $\lim \sup {x_n}$ and $\lim \inf {x_n}$

Define a real sequence recursively by the following equations: $$x_1=0$$ $$x_{2n}=\frac{x_{2n-1}}{2}$$ $$x_{2n+1}=x_{2n}+\frac{1}{2}$$ for each $n \in \mathbb{N}$. Find $\lim \sup {x_n}$ and $\lim ...
1
vote
1answer
47 views

Spherical harmonic related integral as sequence

Consider the following integral: $2\pi\int\limits_0^\pi \sin(x) \cos^2(x) Y_{l,0}(x) Y_{l+2,0}(x) \mathrm{d}x$ Wherein $Y_{l,0}$ are the spherical harmonics for $m=0$, so they are not dependend ...
12
votes
4answers
1k views

How to evaluate the sum $\frac{1}{\sqrt{n}}+\frac{1}{\sqrt{2n}}+\cdots+\frac{1}{\sqrt{n^2}}$ when $n$ grows?

I need help with the following limit $$\lim_{n\to\infty}\sum_{k=1}^n \frac{1}{\sqrt{kn}}$$ Thanks.
1
vote
2answers
80 views

Common mistakes in concluding that series is convergent or not

Assuming $(na_{n})_{n=1}^{\infty}$ convergent to $0$ then $\sum_{n=1}^{\infty}a_{n}$ is convergent, true or false ? wel' I say it is true because: $n$ convergent to infinity and $na_{n}$ is ...
1
vote
3answers
2k views

Bouncing ball geometric sequence question

When a ball falls vertically off a table, it rebounds 75% of its height after each bounce. If it travels a total distance of 490 cm, how high was the table top above the floor? The trouble I am ...
1
vote
2answers
101 views

Find the limit of $\lim_{n \to \infty}\frac{(-1)^{n^2}}{3} + 2 - \frac{1}{(-e)^n}$

I want to find the limes of function: $$\lim_{n \to \infty}\frac{(-1)^{n^2}}{3} + 2 - \frac{1}{(-e)^n}$$ If I break up the statement I get $$\lim_{n \to \infty}\frac{(-1)^{n^2}}{3} = \frac {1}{3}$$ ...
8
votes
6answers
2k views

What is the pattern to this sequence?

$$0, 1, 3, 13, 51, 205$$ More specifically, $$(0,0)\quad(1,1)\quad (2,3)\quad (3,13) \quad(4,51)\quad (5,205)$$ I have tried using the interpolation feature in Grapher.app and Wolfram Alpha, but ...
14
votes
3answers
347 views

How to prove :$\lim_{n\to+\infty}\left(\dfrac{u_{n+1}}{u_1.u_2…u_n}\right)^2=2011$

For sequence $u_n$ satisfing : $$\begin{cases} u_1=\sqrt{2015}\\ u_{n+1}=u_n^2-2\end{cases}$$ How to prove : $$\lim_{n\to+\infty}\left(\dfrac{u_{n+1}}{u_1.u_2...u_n}\right)^2=2011$$
0
votes
1answer
164 views

Series involving gamma functions

How do we represent the summation in the form of elementary functions?? $$ \sum_{n=0}^{\infty}{\frac{x^n \Gamma(n+a)}{n! \Gamma(n+b)}} $$
3
votes
0answers
90 views

Polylogarithm ladders for the tribonacci and n-nacci constants

While reading about polylogarithms, I came across the polylogarithm ladder, $$6\operatorname{Li}_2(1/x)-3\operatorname{Li}_2(1/x^2)-4\operatorname{Li}_2(1/x^3)+\operatorname{Li}_2(1/x^6) = ...
1
vote
4answers
196 views

Geometric Progression with just difference and not ratio

Its given that "In a certain geometric series, the fourth term exceeds the third term by $2$ and exceeds the second term by $5$" and I am supposed to find the third term of this geometric progression. ...
1
vote
2answers
87 views

How to show the convergence?

Let $a_n > 0$ and let $s_n = a_1 +\dots+ a_n$. Prove (a) if $\sum a_n$ converges then $\sum \dfrac{ a_n}{s_n}$ converges. (b) if $\sum a_n$ diverges then $\sum \dfrac{ a_n}{s_n}$ diverges but ...
2
votes
2answers
43 views

expressing $\sum_{k=0}^\infty \sum_{m=0}^k a_k a_m t^{m+k}$ as power series of $t$, $\sum_{n=0}^{\infty}a_n t^n $

How do I change the following type (left one) of series into the one like right one? $$\sum_{k=0}^\infty \sum_{m=0}^k a_k a_m t^{m+k} \to \sum_{n=0}^\infty a_n t^n $$ or $a_{k,m}$ could be the ...
9
votes
1answer
123 views

On Bailey and Crandall's sum for $\sum_{n=0}^\infty \frac{1}{5^{5n}}\left(\frac{5}{5n+2}+\frac{1}{5n+3}\right)$

On page 20 of "On the Random Character of Fundamental Constant Expansions", Bailey and Crandall gave the rather unusual sum, $$u_2 = \sum_{n=0}^\infty ...
2
votes
2answers
82 views

A sequence $(t_{k,n})$ st $\forall n$, $\sum_{k=1}^n t_{k,n}=1$ but $\lim_n t_{k,n} \neq 0$.

I'm looking for an example of a sequence $(t_{k,n})$ such that for all $n$, $$\sum_{k=1}^n t_{k,n}=1$$ but $$\exists k: \ \lim_{n \to \infty}t_{k,n} \neq 0.$$ I've been looking at Toeplitz' lemma for ...
6
votes
2answers
188 views

What should a 21st century Euler attempt?

Euler at the start of his career found the exact sum of the series $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$. My question is: What could a 21st century Euler possibly attempt to solve? Are ...
3
votes
2answers
97 views

Calculate $\lim_{n\to\infty}\left(\prod_{i=3}^n\sec\frac{\pi}{i}\right)$

This question arises from below: Initially you are given a circle $C_2$ with radius $r=1$. You construct the regualr 3-gon (equilateral triangle) with its incircle as $C_2$, and its circumscribed ...