For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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1answer
25 views

How to establish convergence and find limit of the sequence$(n+1)^{1/\ln(n+1)}$

How to establish convergence and find limit of the sequence $(n+1)^{1/\ln(n+1)}$. I know its a stupid question but its kinda urgent so please help me out! Edit 1: It is urgent because I have to ...
0
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0answers
12 views

Absolute Convergence of Infinite Weierstrass Product

I am really stuck on something. I need to show the following: Let $U$ be a domain in $\mathbb{C}$. If $f_n: U \to \mathbb{D}$ are analytic functions satisfying that $\sum |f_n - 1|$ converges ...
2
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2answers
55 views

Taylor expanding $\frac{e^x}{x}$?

How can you taylor expand $$\frac{e^x}{x}$$ Can it be expanded at $x = 0$? Can it be expanded as $x \to 0$?
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0answers
17 views

Can you estimate the difference of primes between numerator and denominator?

Let $p_n$ the nth twin prime, it is $p_n$ is a prime number and $2+p_n$ is also a prime. It is well know that Brun's theorem states (unconditionally) that $$\mathcal{B}=\sum_{n\geq ...
0
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3answers
21 views

How do I decide which convergence criterium to use and what to do if they don't work?

Take $$ \sum_{n=1}^\infty (-1)^nn^{1/n} $$ I just blindly tried (Ratio test) $$ \limsup_{k\to\infty} \left|\frac{a_{k+1}}{a_k}\right| $$ and (Root test) $$ \limsup_{k\to\infty} \sqrt[k]{|a_k|} $$ ...
1
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1answer
12 views

Check if this is the example of $x$ as a limit point of $C$ but ($x_t$) does not converge to $x$.

Let ($x_t$) be a sequence in a metric space, and let $C$ be the range of ($x_t$). I want to give an example in which $x$ is a limit point of $C$ but ($x_t$) does not converge to $x$. Here's my ...
2
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1answer
28 views

How to determine if this musical exercise is valid: will the pattern complete?

I'm hoping that math has an answer to a question arising out of a musical exercise. In music terms, the exercise is: Choose two arpeggios (sets of notes) of equal (or roughly equal) span (number ...
12
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0answers
58 views

Pythagorean triplets of the form $a^2+(a+1)^2=c^2$ and the space between them

I was searching for pythagorean triples where $b=a+1$, and I found using a python program I made the first 10 integer solutions: $0^2+1^2=1^2$ $3^2+4^2=5^2$ $20^2+21^2=29^2$ $119^2+120^2=169^2$ ...
0
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1answer
22 views

Limit Evaluation of a Function in the Complex Field

Given the sequence \begin{equation} z_n=\frac{1}{2n\pi}, \quad n \in \mathbb{N} \end{equation} try to evaluate the following limit: \begin{equation} \lim_{z \to z_n} f(z) \end{equation} where ...
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0answers
27 views

Derive an inequality using Summation by Parts

Can someone help me to derive the following inequality using Summation by Parts? $a_n$ is a decreasing sequence of positive terms. $$\left|\sum_{k=m+1}^{n+p} a_k \sin kx\right| \le ...
2
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1answer
18 views

Sequential compactness of smooth functions

Suppose I have a sequence $u_n$ of smooth functions on the $N$-dimensional reals. If $\|D^{\alpha}u_n\|_{\infty} \leq C_{\alpha}$ for all multi-indices $\alpha$, then is it possible to deduce that ...
2
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0answers
19 views

Is the limit of a Dirichlet series to infinity always equal to the constant term?

Given a sequence of complex numbers $(a_n)$ one defines the corresponding Dirichlet series as $$f(s) = \sum_{n = 1}^\infty \frac{a_n}{n^s}.$$ I know that there is some $x_0 \in \mathbb{R} \cup \{\pm ...
1
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1answer
28 views

A variant of Viète's formula (the 2's replaced by 3's)

I am wondering whether there exists an easy way to evaluate the following infinite product : ...
1
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0answers
26 views

Find minimum value of $n$ for an integer $A$ such that $A=n^x$,where $n>1$ and $x\geq 1$

How can I calculate sum of a series of function $f(A)$ for $A = 2,3,4,5,6...A$ $f(A)=n$ (such that $n$ is minimum integer such that $A=n^x$ where $n>1$ and $x≥1$ and both n and x are integer) ...
0
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1answer
14 views

What does it mean when it says that the sequence of functions $f_n$ decreases monotonically?

Does it mean that if $x>y$ then $f_n(x)<f_n(y)$ for all $n$ or that for all $x$, $f_{n+1}(x)<f_n(x)$?
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2answers
57 views

How to find the nature of this series?

What tests to use for this series: $$\sum_{m=1}^{\infty}(-1)^{m-1}\left(1+\frac{8}{m}\right)^m$$ I've tried alternating test and ratio test but were inconclusive. Can I apply nth term test to the ...
11
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3answers
569 views

Summation of a term to infinity

I read through many tutorials but no one mentioned this explicitly. Is the following conversion valid? $$\sum_{k=0}^\infty \frac{k-1}{2^k} = \lim_{n\to \infty} \sum_{k=0}^n \frac{k-1}{2^k}$$ ...
0
votes
1answer
20 views

Cartesian product to direct sum

I have no idea, how to prove rigorously the corollary from the proposition. I know that i can use the isomorphism $\phi:x_1e_1+...+x_me_m \in \oplus_i^mvect(e_i)\to (x_1e_1,...,x_m e_m) \in ...
3
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2answers
69 views

Can you get a closed-form for $\prod_{p\text{ prime}}\left(\frac{p+1}{p-1}\right)^{\frac{1}{p}}$?

When I use the Taylor expansion series for $$\log(1+x)^{1+x}+\log(1-x)^{1-x}$$ with $x=\frac{1}{p}$, $p$ prime, I believe that I can deduce $$\sum_{p\text{ ...
1
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1answer
49 views

Given the sequence of functions, $f_1(x):=\sin(x)$ and $f_{n+1}(x):=\sin(f_n(x))$, why $|f_n(x)|\leq f_n(1)$?

I'm analyzing this sequence of functions (for $x\in \Bbb R$): $$\begin{align}f_1(x)&:=\sin(x)\\f_{n+1}(x)&:=\sin(f_n(x))\end{align}$$ to show if it converges uniform or pointwise. My book ...
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2answers
57 views

How do I evaluate a series? [on hold]

In this specific example, I don't understand the steps of evaluating this series: \begin{align} &\frac{12}{n}\left(\left[\sum_{i=1}^n-7\right]+\sum_{i=1}^n\left[\frac{-12}{n}\cdot ...
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0answers
29 views

Is there a specific name for these methods of summation?

When calculating summation of series I use these methods ; Ex: Method One $$U_r=\frac{1}{(r-1)r(r+1)(r+2)}$$ $$U_r=\frac{1}{(r-1)r(r+1)(r+2)}\left[\frac{(r+2)-(r-1)}{3}\right]$$ Then ...
12
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3answers
170 views

How to show the divergence of $\sum\limits_{n=1}^\infty\frac{\sin(\sqrt{n})}{\sqrt{n}}$

The 10 standard tests taught in class are: 1) $n^{th}$ term test for divergence.(Not applicable: $\lim =0$). 2) Geometric Series(Not applicable). 3) Telescoping Series(Not applicable) 4) Integral ...
7
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3answers
87 views

What is the sum of all the natural numbers between $500$ and $1000$.

What is the sum of all the natural numbers between $500$ and $1000$ (extremes included) that are multiples of $2$ but not of $7$?
2
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1answer
18 views

Splitting a sum to find a closed form of $\sum_{n=2}^\infty\frac{n}{n-1}x^{n-1}$

Find a closed form for $$S = \sum_{n=2}^\infty\frac{n}{n-1}x^{n-1}$$ My solution The radius of convergence is $R=1$ and the series does not converge in $\pm 1$. Rewrite the sum as ...
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2answers
29 views

What is the correct notation for every nth term in a sequence?

How do I denote every nth term in a sequence? For example, if sequence $C$ contains: $C = \{ 2, 5, 3, 6, 4, 5, ...\}$ And sequence $Q$ contains every 4th term in C: $Q = \{C_{4}, C_{8}, ...
5
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1answer
65 views

Find the limit of this sequence

Suppose $$ f_n(x)=\sum_{k=1}^n \frac{\cos(kx)}{k}, $$ and let $a_n=\min_{x \in [0,\pi/2]} f_n(x)$, find $\lim_{n \to\infty} a_n$. I wrote a program and found that the $\arg\min_{x \in [0,\pi/2]} ...
1
vote
1answer
19 views

Why is the sequence $u_N = \inf\{s_n : n \gt N\}$ increasing?

A question in my book I am studying says to let $s_n$ and $t_n$ be sequences and suppose there exists $N_0$ such that $s_n \le t_n$ for all $n \gt N_0$. Show $\lim \inf s_n \le \lim \inf t_n$ and ...
0
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2answers
29 views

Remainder value from $0$ to $9999$

I was trying to find how many numbers from $0$ to $9999$ that have the remainder value of $23$. I tried writing a program to help me solve that but it got me nowhere. There has to be a simpler way to ...
0
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1answer
24 views

Construction of sequence from convergent susbsequences

Is it possible to construct the following? A sequence that contains subsequences converging to every point in the infinite set $\{{1, 1/2, 1/3, 1/4, 1/5, ...}\}$ and no subsequences converging to ...
0
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0answers
30 views

Is there a divergent sequence such that for every n in N it is possible to find n consecutive ones somewhere in the sequence

I was asked to create, if possible, a divergent sequence such that for every $n$ in $N$, it is possible to find '$n$' consecutive ones somewhere in the sequence. I came up with the sequence: $\{1, ...
0
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1answer
57 views

Sum of the series $2+5+14+41+…$ [on hold]

How can we find sum of the following series upto $n$ terms? $S=2+5+14+41+.....$ As I can see, pattern here is: $5=3(2)-1$ $14=3(5)-1$ $41$=3(14)-1 Is it possible to find sum of $n$ terms?
2
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5answers
63 views

Are there two different unbounded sequences such that if you subtract them they converge to $0$?

I'm having a hard time coming up with two unbounded sequences where their difference yields $0$ when $n\rightarrow\infty$. Any ideas?
0
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1answer
66 views

Completing the sequence, is answer $98$ or $99$? [on hold]

In this sequence, what is the formula or series being followed? I framed a formula: $a^2 - ((a-b) * 2)$ Which derives to $98$. But it also appears like they are multiplying a and b and adding a odd ...
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0answers
22 views

Difference Equations and Discrete Integral and Derivative

So I'm trying to learn difference equations, and the book that I'm using defines the following: The discrete derivative of a function $a_n$ of the integers is defined as: $$ D a_n = a_{n+1} - a_n $$ ...
6
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1answer
84 views

Is there a series to show $22\pi^4>2143\,$?

This extends this post. I. For $\pi^3$: $$\pi^6-31^2 =\sum_{k=0}^\infty\left(-\frac{63}{(2k+2)^6}+\frac{31^2}{(2k+3)^6}\right) =\sum_{k=0}^\infty P_1(k)\tag1$$ As pointed out by J. Lafont, when ...
0
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1answer
14 views

Uniformly convergent in a set implies uniformly convergent in the set closure, too.

Let $f_n$:$X\rightarrow \mathbb{R}$ be a sequence of functions uniformly convergent in $X\subseteq \mathbb{R}$ . Suppose that each $f_n$ is continuous in the closure of $X$. Then $f_n$ is also ...
0
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1answer
30 views

On a recursive sequence exercise $a_{n+2} = \frac{4 + 3a_n}{3 + 2a_n}$.

As part of an exercise I am given a sequence defined by $a_1 = 1$ and $$a_{n+1} = 1 + \frac{1}{1 + a_n}$$ I have noticed that the even sequence is decreasing and I want to prove this, the even ...
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4answers
64 views

Explicit formula for $e_k = 4e_{k-1} + 5$

The sequence looks like this: $e_0 = 2$ $e_1 = 4(e_{1-1}) + 5 = 13$ $e_2 = 4(e_{2-1}) + 5 = 57$ $e_3 = 4(e_{3-1}) + 5 = 233$ $e_4 = 4(e_{4-1}) + 5 = 937$ How would I go about finding the ...
0
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1answer
40 views

Finding a closed formula for: $1\cdot2\cdot3+2\cdot3\cdot4+…+(n-2)\cdot(n-1)\cdot(n)$ [duplicate]

As I calculated the sum of the serie above doesn't exist(sum doesn't converge). How can I prove it using the double computing(combinatorical method)?
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2answers
44 views

What type of series is $A_1 + A_2 n + A_3 \frac{n(n+1)}{2}$

I am solving a coding problem and I break it down to a point where I get a series like this: $$A_1 + A_2 n + A_3 \frac{n(n+1)}{2} + A_4 \frac{n(n+1)(n+2)}{2\cdot 3} + A_5 ...
2
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2answers
85 views

Prove that if $\sum a_n$ converges, then $na_n \to 0$. [duplicate]

Let $a_n$ be a decreasing sequence of nonnegative real numbers. Prove that if $\sum a_n$ converges, then $na_n \to 0$. Hint: use that $n\, a_{2n} \le a_{n+1}+\cdots + a_{2n}$ I couldn't ...
0
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1answer
23 views

Show the convergence of a sequence using a convergent sequence in a metric space

Let $X$ a metric space with metric $d$ and let $P \subset X$ not empty. Let $f :X \to \mathbb{R}$ a function and define $f(x) = \inf\{d(x,y): y \in P\}$. Let $(x_n)_{n \in \mathbb{N}}$ a convergent ...
0
votes
1answer
44 views

Way to calculate cumulative sum of a sequence m times without summing all elements m times.

Suppose we have a sequence 2 1 3 1 Now , I want calculate it's cumulative sum m times and determine the element at position x in the sequence. Lets's say I want to perform cumulative sum operations ...
1
vote
1answer
25 views

Property of a sequence being an enumeration of the rationals.

Let $(r_n)$ be an enumeration of the rationals and $x\in\mathbb{R}$. Is it possible to find out whether the set $\left\{n\in\mathbb{N}:\left|x-r_n\right|<\frac{1}{2^n}\right\}$ is finite or ...
0
votes
0answers
16 views

Approximate ratio with a small fraction so that numerator multiplied by denominator give enough rectangular area?

I would like to layout given number of objects (like plots) into rectangular area (like computer operating system window on screen). I would like to calculate the width and height of the window (in ...
5
votes
1answer
51 views

Express a real number as a product

Hi guys if I have a number $x \in [1,2)$ is it possible to express such number as: $$x = \prod_{j=0}^{+\infty} (1 + \alpha_j 2^{-j})$$ where each $\alpha_j \in \left\{-1,0,1\right\}$? If yes, how ...
1
vote
1answer
44 views

Power series converging at the convergence radius

Let $f(z)=\sum a_nz^n$ be a power series of radius $R$. By Abel' radial theorem, if $f(R)$ converges then $f$ is continuous over real numbers at $R^-$. I had some questions on how that can be ...
2
votes
2answers
100 views

Is there any better way to find n! without just multiplying all the numbers till n? [duplicate]

I was solving some permutation, combination problems where we know that we are to use factorial. So I was thinking if there was any shorter way, maybe a formula, than multiplying all the numbers till ...
2
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0answers
20 views

dimension and base of arithmetic sequence

Arithmetic sequence is a vector space. But how to find a dimension of it. I try this: arithmetic sequence: $a_1,a_2,...,a_n={a_1,(a_1+d),(a_1+2d),...,(a_n+(n-1)d)}$ With only $a_1$ and $d=(a_1-a_2$) ...