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

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0answers
15 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
43 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 ...
0
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2answers
49 views

How do I evaluate a series?

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 ...
0
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0answers
19 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 ...
10
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2answers
134 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
75 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
16 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 ...
1
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2answers
28 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
61 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
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1answer
18 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
28 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
22 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
20 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
55 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?
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5answers
53 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
65 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 ...
1
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0answers
20 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: $$ D a_n = a_{n+1} - a_n $$ ...
4
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0answers
47 views

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

This extends this post. I. For $\pi^3$: $$\pi^6-31^2 =\sum_{k=0}^\infty\left(-\frac{63}{2^6(k+1)^6}+\frac{31^2}{(2k+3)^6}\right) =\sum_{k=0}^\infty P_1(k)\tag1$$ As pointed out by J. Lafont, ...
0
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1answer
12 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
29 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
59 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
40 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
votes
2answers
81 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
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1answer
35 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
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1answer
24 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
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0answers
14 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
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1answer
50 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
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1answer
40 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
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2answers
97 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 $d=(a_1-a_2$) ...
0
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2answers
26 views

Different versions of Bolzano Weierstrass Theorem and their relationships.

Which one is the Bolzano Weirerstrass Theorem? Theorem 1. Every bounded sequence in $\mathbb{R}^n$ has a convergent subsequence. OR Theorem 2. Every sequence of real numbers has a monotonic ...
0
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3answers
50 views

series $\sum_{n=0}^{\infty}9^{n}z^{2n}$

have to calculate the ratio of the serie in the title. So using the ratio test criteria I find that $\frac{9^{n+1}}{9^{n}}=9$ and so that $R=\frac{1}{9}$. My professor's result is $\frac{1}{3}$ ...
0
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1answer
21 views

Convergence of $f_n = \frac{x}{3-5n|x|}$

Study the convergence of the sequence $$f_n = \frac{x}{3-5n|x|}$$ The domain of $f_n$ is $\operatorname{dom} f_n = \mathbb R \backslash \{\pm\frac3{5n}\}$ and $$\lim_{n \to +\infty} f_n = 0$$ ...
0
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0answers
18 views

What is the maximum number of times that absolute value of neighboring difference is larger than a threshold?

Suppose I have a sequence of data $x_1$, $x_2$, ..., $x_N$. Suppose for a particular value $X$, and for a particular interval $m$, the number of times that $|x_{i+m}-x_i|>X$ ($1\le i\le N-m$) is ...
0
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1answer
39 views

Sum and radius of $\sum_{n=0}^{\infty}(\cos(5n)+i \sin(5n))z^{n}$

I calculated radius and sum of the series in the title. First i converted that in the exponential form: $\sum_{n=0}^{\infty}e^{i5n}z^{n}$ then I applied the ratio test and i got a value of $e^{i5}$ ...
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0answers
37 views

What does a variable superscript above a set mean?

I'm not entirely sure I've worded this correctly. An example of what I mean is... $$U = \{0,1\}^n$$ What is the meaning of the superscript?
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1answer
35 views

I roll a fair die repeatedly until I get $6$, what is the probability that neither $1$ nor $2$ occurs before $6$ appears.

I roll fair a die repeatedly until I get $6$, what is the probability that neither $1$ nor $2$ occurs before $6$ appears. Not sure how to go about this.
3
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2answers
47 views

Show that the integer nearest to $\frac{n!}{e}$ $(n \geq 2)$ is divisible by $n − 1$ but not by $n$.

Show that the integer nearest to $\frac{n!}{e}$ $(n \geq 2)$ is divisible by $n − 1$ but not by $n$. I am still trying to improve my basic math skills but on this one i did not get far. Taylor ...
1
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1answer
22 views

What is the radius of convergence of following series?

suppose that $\sum b_n$ is conditionally convergent but not absolutely convergent. What is the radius of convergence of the following power series $p(x)=\sum b_nx^n$?
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1answer
70 views

Does this prove the sequence $5+(-1)^n$ does not have a limit?

The question is "Consider the sequence $s_n=5+(-1)^n$. Prove that this sequence does not have a limit". My professor in class proved this by choosing $n_1$ to be even, $n_2$ to be odd, and ...
0
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1answer
31 views

$S_n = \sum_{k=1}^{n} (\sqrt{1 + \frac{k}{n^2}} - 1)$ Show that $\lim_{n \rightarrow \infty}S_n = \frac{1}{4}$

Show that $\lim_{n \rightarrow \infty}S_n = \frac{1}{4}$ $$S_n = \sum_{k=1}^{n}\left(\sqrt{1 + \frac{k}{n^2}} - 1\right)$$ $$\sum_{k=1}^{n}\left(\sqrt{1 + \frac{k}{n^2}} - 1\right) < ...
2
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3answers
46 views

Proof with Fibonacci Sequence

I was working on my homework assignment for one of my classes and I have come across a proof question that my classmates and I are finding difficult to answer. The problem is asking us to prove that ...
1
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2answers
44 views

For a series, can you always find a subseries whose sum is smaller in magnitude?

Let's say you got a series $a_n$, such that $\sum^\infty_{n=0}a_n=L>0$. For any $0 \lt K \le L$, can you always find a subsequence $b_n$ of $a_n$, such that $\sum^\infty_{n=0}b_n=K$? If not always, ...
6
votes
6answers
79 views

Closed form of $\sum\limits_{n=1}^\infty \frac{4^n(x+4)^{2n}}n$

Let $$S(x) = \sum_{n=1}^\infty \frac{4^n(x+4)^{2n}}n$$ 1. Find the radius of convergence. 2. Calculate $S(x)$. 3. Find $S^{(n)}(x)$ without computing the derivatives of $S(x)$. From the ...
0
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0answers
16 views

Odd perfect numbers and $\sum_{\substack{D\mid 2n,D<\sqrt{2n}}}(D+\frac{2n}{D})$

It is known that if $m>1$ isn't a perfect square integer (isn't a square number) then the sum of divisor function can be written as $$\sigma(m)=\sum_{\substack{d\mid ...
4
votes
4answers
96 views

Evaluation of $\,\displaystyle \lim_{n\rightarrow \infty}\sum_{k=1}^n\sin \left(\frac{n}{n^2+k^2}\right)$

Evaluation of $\displaystyle \lim_{n\rightarrow \infty}\sin \left(\frac{n}{n^2+1}\right)+\sin \left(\frac{n}{n^2+2^2}\right)+\cdots+\sin \left(\frac{n}{n^2+n^2}\right)$ $\bf{My Try::}$ We can ...
3
votes
5answers
92 views

Give examples showing why $0\cdot \infty$, $\infty/\infty$, and $0/0$ are meaningless

Assuming arithmetic operations on $\overline{\mathbb{C}}$ (that's the extended complex plane) are defined via arithmetic operations on the corresponding sequences, I need to give examples showing why ...
0
votes
0answers
15 views

Find the interval of convergence of the given series and study its nature on its edges

Firstly, I'm not sure if the title is written correctly, because I am not a fluent English speaker, but I hope you understand what I'm talking about (Any edits would be welcome). However, let's move ...