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

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-1
votes
0answers
24 views

Using Cauchy's convergence test on a specific series

This is my series: Series WolframAlpha tells me that this series converges. Here's what I get instead with the Cauchy convergence test: My solution What am I doing wrong exactly?
0
votes
1answer
18 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
21 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
0answers
10 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
11 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
42 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
23 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
93 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
votes
0answers
17 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
votes
2answers
22 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
votes
3answers
39 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
votes
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 $$\lim_{n \to +\infty} f_n = 0$$ ...
0
votes
0answers
11 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
votes
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 $e^{i5}$ ...
0
votes
0answers
35 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?
1
vote
1answer
31 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
votes
2answers
42 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
vote
1answer
19 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 $p(x)=\sum b_nx^n$?
0
votes
1answer
63 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
votes
1answer
29 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
votes
3answers
44 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
vote
2answers
35 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
75 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
votes
0answers
14 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
87 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 ...
0
votes
1answer
29 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 ...
0
votes
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 ...
1
vote
1answer
71 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
votes
0answers
121 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
18 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
31 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
votes
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?
-1
votes
1answer
10 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
votes
1answer
26 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 ...
1
vote
2answers
23 views

Forming a sequence from a Cauchy Sequence

Let $(a_{n})$ be a Cauchy sequence. Is $c_{n} = (-1)^{n}a_{n}$ also a Cauchy sequence?
5
votes
1answer
69 views

If $\sum a_n$and$\sum b_n$diverge, can$\sum \min\{a_n,b_n\}$converge? [duplicate]

Do there exist sequences $\{a_n\}$ and $\{b_n\}$ satisfying all of the following properties? $a_n>0$ and $b_n>0$ $\{a_n\}$ and $\{b_n\}$ are both decreasing $\sum a_n$ and $\sum b_n$ both ...
4
votes
2answers
66 views

Finding limit via Sandwich Theorem: $\lim_{n\to\infty} n\sum_{n+1}^{2n} \frac{1}{i^2}$

Question: Use the Sandwich Theorem to find $$\lim_{n\to ∞} n\sum_{n+1}^{2n} \frac{1}{i^2}$$ Appreciate any guidance.
3
votes
1answer
32 views

Can you prove this identity involving the divisor sum function?

Let $$\sigma(n)=\sum_{d\mid n}d$$ the sum of divisor function. I've deduced, but I don't know how prove directly $$\sigma(n)=\sum_{m=1}^{n}\sum_{k=1}^{m}(-1)^{nk}\cos\left(\frac{\pi ...
2
votes
2answers
45 views

Limit of the sequence $\lim_\limits{n\to\infty}\sin(2\pi(n^2+n^{1/2})^{1/2})$.

I have tried to solve this limit : $\lim_\limits{n\to\infty}\sin(2\pi(n^2+n^{1/2})^{1/2})$. Where n $\in\mathbb{N}$. I have understood that the limit exists and goes to 0 if the argument becomes ...
1
vote
0answers
31 views

Can a sum of trigonometric functions equal a constant for all inputs?

Let $r_1,...,r_n$ and $\phi_1,...\phi_n$ be real numbers. Consider the following sum: $S=\sum\limits_{k=1}^{n}r_k\sin(\phi_k+k\alpha)$ Suppose $S$ is constant for all $\alpha \in R$. Does it ...
1
vote
1answer
34 views

Finding limit of $\frac{a_n^3+5n}{a_n^2+n}$ for $(a_n)$ bounded.

Suppose that the sequence $(a_n)_{n \in \mathbb{N}}$ is bounded. Prove that the sequence $(c_n)_{n \in \mathbb{N}}$ defined by $$ c_n = \frac{a_n^3+5n}{a_n^2+n} $$ is convergent and find its ...
-2
votes
1answer
29 views

To check whether series S and T are convergent or not [on hold]

To check whether series S and T are convergent or not . I applied ratio test for series S and found it to be convergent but i do not know about series T. Thanks
1
vote
4answers
54 views

Is the sequence $\sqrt{1+\frac{1}{n^2}}$ increasing or decreasing?

Is the sequence $\sqrt{1+\frac{1}{n^2}}$ increasing or decreasing? I simplified it to $\frac{\sqrt{n^2+1}}{n}$, and I tried $a_{n+1}-a_n$ and $\frac{a_{n+1}}{a_n}$, but neither seem to work, how ...
0
votes
0answers
59 views

Find the Summation of Summation [on hold]

An Array A consisting of N integers .We perform the following operation M times: for i = 2 to N: Ai = Ai + Ai-1 We have to find xth element of the array ...
0
votes
1answer
30 views

Epsilon delta proofs of theorems of continuity

Can anyone suggest a book which contains epsilon delta prooves for properties and theorems of continuity rather than sequential proofs.
2
votes
1answer
30 views

Does this conjecture in 1-d real analysis seem reasonable

Hello I am currently trying to prove a result and I have basically whittled it down to showing the following is true. Let $I\subset\mathbb{R}$ be an interval and fix $\alpha>1$ real number. Fix ...
-3
votes
2answers
52 views

Is the sequence $s_n= \frac{\sin\frac\pi2}{1\cdot 2}+\frac{\sin\frac\pi{2^2}}{2\cdot 3} + \dots + \frac{\sin\frac\pi{2^n}}{n\cdot(n+1)}$ convergent? [on hold]

Which of following is correct? I think option D. Not sure though $$s_n= \frac{\sin\frac\pi2}{1\cdot 2}+\frac{\sin\frac\pi{2^2}}{2\cdot 3} + \dots + \frac{\sin\frac\pi{2^n}}{n\cdot(n+1)}$$ is ...
1
vote
1answer
26 views

Harmonic progression sum

http://www.mathalino.com/reviewer/algebra/arithmetic-geometric-and-harmonic-progressions Please go to this link and see how they tell you to find the sum of harmonic prgression. However I am sure it ...