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

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7 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 ...
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2answers
20 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 ...
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1answer
18 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 ...
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0answers
17 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, ...
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1answer
48 views

Sum of the series $2+5+14+41+…$

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
50 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?
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1answer
57 views

Completing the sequence, is answer $98$ or $99$?

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
17 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: $$ D a_n = a_{n+1} - a_n $$ ...
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0answers
42 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, ...
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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 ...
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1answer
24 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 ...
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1answer
39 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 ...
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2answers
79 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 ...
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1answer
21 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 ...
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1answer
26 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 ...
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0answers
16 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 ...
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0answers
13 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
46 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 ...
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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
95 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 $d=(a_1-a_2$) ...
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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 ...
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3answers
41 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 $\frac{1}{3}$ ...
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1answer
20 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 $$\lim_{n \to +\infty} f_n = 0$$ ...
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0answers
17 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
37 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
36 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
33 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.
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2answers
46 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 ...
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1answer
21 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
68 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 ...
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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
45 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
42 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, ...
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6answers
76 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 ...
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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 ...
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4answers
92 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
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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 ...
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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 ...
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1answer
30 views

Determine whether a property possessed by every term in a convergent sequence is necessarily inherited by the limit.

I'm having difficulty coming up with actual sequences that have the properties below. I've included my thoughts on the questions below. Assume that $(a_n)\rightarrow a$. If every $a_n$ is an upper ...
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1answer
11 views

Limits of the derangements proportion within the permutations of the set $[1,n]$

Let be $D_n$ the number of derangements of a set of $n$ elements, by convention we have $D_0=1$ Ifound that $D_n=n!\sum\limits_{k=0}^{n}\frac{(-1)^k}{k!}$ For all $n\in \mathbb{N*}$, we write ...
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1answer
74 views

Recurrence relation $a_{n+1}a_{n-1} = 1 + a_n$ [on hold]

Consider the recurrence relation: $a_{n+1}a_{n-1} = 1 + a_n$ with initial values $a_1=x$ and $a_2=y$. Is this an example of a homogeneous equation or just a linear one? In any case does anyone have ...
5
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0answers
137 views

A monotonically increasing series for $\pi^6-961$ to prove $\pi^3>31$

This question is motivated by Why is $\pi$ so close to $3$?, Why is $\pi^2$ so close to $10$? and Proving $\pi^3 \gt 31$. I. $\pi$ and $\pi^2$ There are series with all terms positive for ...
1
vote
1answer
19 views

Divergence of squared sum of Chebyshev Polynomials $\equiv L+R$ has empty point spectrum

The Chebyshev Polynomials of the second kind $U_n$ are the solutions of the differential equation $$(1-x^2)U_n''(x)-3xU_n'(x)+n(n+2)U_n(x)=0$$ Alternatively they are defined inductively: $$U_0(x)=1 ...
2
votes
1answer
32 views

Finding the maximum value of a divergent series [on hold]

I came across this divergent sum- $$\sum_{n=1}^\infty\frac{1}{n+1}$$ Now,a divergent sum does not a limit.So is it possible to get a maximum value for the sum or more specifically prove that ...
0
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1answer
24 views

how to write as geometric series $\dfrac{A(3s-5)}{(s-3)(3s-5)}+\dfrac{B(s-3)}{(3s-5)(s-3)}$ [on hold]

How would I write $\dfrac{A(3s-5)}{(s-3)(3s-5)}+\dfrac{B(s-3)}{(3s-5)(s-3)}$ as a sum of geometric series?
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1answer
11 views

Given common terms (and their position) between an arithmetic and geometric sequences, find the common ratio. [on hold]

The fourth, seventh and sixteenth terms of an arithmetic sequence also form consecutive terms of a geometric sequence. Find the common ratio of the geometric sequence
0
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1answer
33 views

Odd and Even Fourier Series Extension of $f(x)=x$ on $[0,\pi]$

I'm confused on finding the odd and even extensions of $f(x) = x$ on $[0,\pi]$. I know the general forms and how to find the co-efficients, but for the sin series, $f(0)$ =/= $f(\pi)$, so then I only ...