Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: recurrence relations, convergence tests, identifying sequences, identifying terms. For questions on finite sums, use the (summation) tag instead.

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Let $\sum_{n=0}^{\infty} a_n = M$ be a convergent series with $a_n \ge 0 \in \mathbb R$. Non-contradiction proof: $s_k \le M$ for $k \in \mathbb N$.

Let $\sum_{n=0}^{\infty} a_n = M$ be a convergent series with $a_n \ge 0 \in \mathbb R$. I want to prove $s_k \le M$ for $k \in \mathbb N$. I could do a proof by contradiction as follows: Suppose $...
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19 views

number of indices differences

I have a counting problem below: Let $n>2$ be integer and $p>0$ be a real number. For all $1\leq i<j\leq n$, suppose $a_{ij}$'s and $b_k$'s satisfy $a_{ij}=b_{j-i}=p^{j-i}$ $b_{j-i}=b_{n-(j-...
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112 views

What was Euler's misconception about functions and infinite series?

I just read this on Strichartz' The way of Analysis: [...] Euler - the leading mathematician of the eighteenth - developed all techniques needed for the study of Fourier series, but he never ...
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1answer
419 views

Frequency analysis/discrete uniform distribution in multiple choice tests

I may be using the wrong terms in the title but I read that if something is random then each character will occur an equal amounts of times. I read this when reading about the One-Time Pad cipher, ...
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2answers
90 views

representing a recursive difference equation of two variables into one variable equation

suppose the following recursive difference equation ($t$ is time): $$x_t = \frac{a}{1+a}x_{t-1} + \frac{1}{1+a}x_{t+1}$$ where $0<a<1$ is assumed and all values of $a$ at past times are ...
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1answer
29 views

Validating a proposition

Proposition: For all $k,n\in\mathbb{Z^+}$ $s.t$ $n\lt4$ $2{n\choose n}+{n\choose n-1}+...+{n\choose k-(n-2)}=2^n$ for $1\le k\le n-1.$ I understand that this proposition is invalid, so are there ...
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1answer
158 views

Sequence with a fixed last element Notation

I was trying to write a sequence of two different elements (that always appear in order) with a fixed last element, for an example: $A_1, B_1, A_2, B_2, A_3, B_3, A_4$. I'm not sure which would be the ...
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1answer
168 views

Hypothesis testing for equivalence of two arrangements

I have two arrangements(i.e. permutations) of numbers. First one is the target/real arrangement. Second, is the observed arrangement. e.g. Target := 1,2,3,4,5,6,7 Observed := 4,1,7,3,2,5,...
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53 views

Is something wrong in this proof?

Show that if $\sum a_nx^n$ has convergence radio $R$ and $\limsup |a_n| > 0 $, then $R\leq 1$. Proof: Suppose that $\sum a_nx^n$ has convergence radio $R$ and $\limsup |a_n|=\alpha > 0$. ...
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2answers
238 views

Proving that $\sum_j x^j$ is differentiable $(-1,1)$

I'm really struggling with understanding how to apply the Weierstrass M-test, and so some hints on this question would be much appreciated: First I want to prove that $\sum_{j = 0}^\infty x^j$ is ...
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1answer
209 views

lim sup of two sequences

Let $(a_n)_{n \in\ \mathbb{N}}$ a bounded sequence in $\mathbb{R}$. For $n \in \mathbb{N}$ let $$v_n=\sup\{a_k; ~k \geq n\},\quad u_n=\inf\{a_k; ~k \geq n\},\quad s_n=\sup\{|a_k-a_l|; ~k,l \geq n\}$...
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102 views

examine the convergence of the series

$$a_n=\frac{1}{n*10^{log(logn)}}$$ I have to examine $\sum{a_n}$ so I used Cauchy's condensation test and I got: $$b_n=\frac{2^n}{2^n*10^{log(log2^n)}}$$so $$b_n=\frac{2^n}{2^n*10^{log(log2)^n}}$$ ...
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1answer
21 views

Radius of convergence of a series (text problem)

the series: $\sum_{n=1}^\infty a_nx^n$ where $(a_n)_n$ is a limited sequence with $L((a_n)_n) \subseteq \mathbb{R}\backslash \{0\}$ My main problem is to get to something to work with. I dont know ...
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70 views

Recurrence in two variables

Anyone know how to solve the following recurrence relation in two variables: $$ f(x,y) = b f(x-1,y) + c f(y,x-1), \qquad \begin{cases}f(x,0) = b^{(x-1)} \\ f(0,y) = 0 \end{cases} $$ (Note: repost of ...
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1answer
71 views

'ϵ-δ' proof for the following sequence

I need help writing a formal 'ϵ-δ' proof for the following sequence: $$ \lim_{n\to \infty}(n+2)^2 \sin(1/n)=\infty $$ Thanks in advance.
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54 views

Check definition-Decreasing sequence

Is the following math definition of a decreasing sequence from a certain range correct? $\exists n_0. !n. n \ge n_0 \Rightarrow f(n+1) <= f(n)$ I mean by "from a certain range", that when $n \ge$ ...
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33 views

Error estimation help

I'm supposed to find a Taylor polynomal of the $n^{\text{th}}$ degree, where $x = a$, and estimate the error for the given interval. The problem I'm given is: $$f(x) = \sqrt{x}, a = 4, n = 2, 4 \leq x ...
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81 views

simple undergraduate series quesiton

consider $ \displaystyle \sum_{n=1}^\infty (-1)^{n-1}a_n $ where $ (a_n) $ is a monotone decreasing sequence of nonnegative numbers with $ a_n \rightarrow 0 $ by the alternating series test, series ...
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81 views

Finding interval convergence $\sum\frac{x^k}{k^k}$

How would I find the interval of convergence for $\sum\frac{x^k}{k^k}$ I did the ratio test. $\frac{x^{k+1}}{(k+1)^{(k+1)}}$*$\frac{k^k}{x^k}$ $\frac{x k^k}{(k+1)(k+1)^k}$ I got $k\rightarrow\...
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30 views

How many combinations is possible this two particular series

thanks in advance answering this question. Suppose i have two series, with series 1 $ [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] $ and series $ 2 , [x,x,x,x,x,x,x,x,x,x,x,x,x,x] $ with length=$ 14 $ ...
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42 views

Is $f(t)=2t\sin(1/t)-\cos(1/t)$ for $t \neq 0$ and $f(t)=0$ for $t=0$ differentiable?

Is the function $f$ defined as: $f(t)=2tsin(1/t)-cos(1/t)$ for $t \neq 0$ and $f(t)=0$ for $t=0$ $t \in R$ differentiable? In my opinion it's not because taking a sequences we can easily check ...
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46 views

Geometric series - regarding the exponents of the factor r

When giving a proof for the formula of a geometric sum, the following sequence is used $$S = a + ar + ar^2 + ... + ar^{n-1},$$ Why does the last term go to $n-1$ and not just $n$? I wrote the last ...
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300 views

Asymptotic (in)dependence of a maximum of an i.i.d. sequence of Gaussian random variables on a single random variable in this sequence

Suppose that I have a sequence of $n$ i.i.d. standard Gaussian random variables $X_1,\ldots,X_n$ where $X_i\sim\mathcal{N}(0,1)$. Denote the maximum of this sequence by $M_n=\max(X_1,\ldots,X_n)$. I ...
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2answers
97 views

binomial theorem formula with all coefficients being 1

I happened to wonder that for the binomial expansion formula, if all the coefficients for each term is just 1 as opposed factorials, can we still write a nice formula without the sum sign? I really ...
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1answer
59 views

Sequence in span with disjoints supports has (?) block subsequence.

Assume $b_k \in <e_i, \text{with coefficients} \ a_i^k \geq 0>$ is a sequence and $a_i^k$ have disjoint supports(support is the set where $a_i^k \neq 0$). Is there a way to prove or disprove ...
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28 views

Sequence values/ Discrete Mathematics

$H_0=5, H_2=5$ Is $H$ nondecreasing? I really don’t understand how to prove/answer this mathematically if both values are the same?
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28 views

String satisfying the condition

Given $N$, $A_0$, $B_0$, $L_0$, $A_1$, $B_1$ and $L_1$, find a sequence S consisting only of characters '$0$' and '$1$'(a total of N characters) such that: The number of '$0$'s in any consecutive ...
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1answer
131 views

What is this the name of this idea? (combinatorics)

The problem: There are three screws, each one a different type {Phillips, Robinson, Slotted}. There are three sets of screwdrivers, each set corresponds to a type of screw. There are no two ...
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70 views

Sum Series Simplication

I'm trying to simplify the following formula: $$\sum_{1\leq z < z^\prime < y\leq n} \frac{k^3}{z z^\prime y} (1 + \frac{k}{x + 1}) (1 + \frac{k}{x + 2}) \ldots (1 + \frac{k}{z - 1})(1 + \frac{k}...
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1answer
130 views

Geometric sequences interest question

Winston invests a sum of money at 6% per annum. How many years does it take him to double his money? I let the initial sum of money be $£a$. Then at the end of the first year, he has $£1.06a$ since ...
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45 views

Convergence of Series over Multi-integers

Given a dimension $n\in\mathbb{N}$, is there an easy way to say why the following sum over the multi-integers should converge? (By $\|k\|$ I mean the Euclidian distance on $\mathbb{R}^n$.) $$\...
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1answer
45 views

about function series

Good evening, everyone, Could anyone please tell me how to check if the series $\sum_{k\geq 2}\dfrac{1}{k^4+x}$ is greater than $C\sum_{k\geq 2}\dfrac{1}{k^2+x}$ where $C$ is independent of the ...
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1answer
29 views

Determining a sequence generation variable

Let $$ \begin{align} r_1 &= c + d + \frac{d}{2} + \frac{d}{4} + \dots + \frac{d}{2^n}\\ r_2 &= c + d + \frac{d}{2} + \frac{d}{4} + \dots + \frac{d}{2^n} + \frac{d}{2^{n+1}} \\ r_3 &= c + ...
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75 views

Help with creating equation

I wanted to know how many times the number $11$ appears at least once in any decimal expansion. For instance some decimal expansions under $10^6$ could be: $341011$, $511993$, $118$, $1101100$ all ...
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1answer
46 views

Real analysis: convergence question

I found the closed formula for the sum of 1/( k^2+3k+2) from 1 to infinity which is 1/2 - 1/(n+2). Could you first check whether this is right. If so, how to find the sum of this series? Is it just ...
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1answer
67 views

Main source of information for digamma function?

Is there a great source of information for the digamma function $\psi(x)=\frac{d}{dx}\log\Gamma(x)$, $x>0,$ that is not already on the Wikipedia page (http://en.wikipedia.org/wiki/Digamma_function)?...
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1answer
65 views

Proving a Space of Real Valued sequences is Banach.

Theorem: A normed vector space $(V,||\circ||)$ is a banach space if and only if for every sequence $x_n$ in $V$ with the property that $\sum ||x_n||<\infty$ we have $\sum x_n < \infty$. ...
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94 views

convergent and bounded sequence

Denote $E$ the set of all real sequences $\{a_n\}$ such that $|a_n| ≤ 1$ for every positive integer $n$, $\ell^1$ be the set of all real sequences $\{a_n\}$ such that $\sum a_n$ converges absolutely, $...
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113 views

Convergence of $\sum_{n=0}^{\infty} \frac{(-1)^{n +1}}{2n+1}$

In a textbook it is claimed that $\displaystyle \sum_{n=0}^{\infty} \frac{(-1)^{n +1}}{2n+1}=\frac{-\pi}{4}$ converges. How would I calculate to what it converges? Is there some formula with which ...
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31 views

Quick simplification strategy for $\binom{3}{2}p^2(1-p) \le \sum_{k=3}^{5}\binom{5}{k}p^k(1-p)^{n-k}$

What is a quick simplification strategy to solve the following expression for $p$ by hand? (or less preferably, by a TI83/86 calculator). $$\binom{3}{2}p^2(1-p) \le \sum_{k=3}^{5}\binom{5}{k}p^k(1-p)...
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1answer
382 views

Finding moment generating function of a discrete random variable (series)

So the question is find the moment generation function for a random variable X with pmf $f_X(x)={{r+x-1} \choose x} p^r (1-p)^x ,x=0,1,2,\ldots, 0<p<1,\mbox{ and }r\in\mathbb{Z}^+$. So it's a ...
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1answer
111 views

Base conversion using geometric series

I'm working on converting numbers in various bases and one question asks to convert $.2525...$ from decimal to octal. I know that the answer is $1/3$ and that it is necessary to use the infinite ...
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57 views

Working with infinite sequences in $\ell^2(\mathbb{Z})$

Let $\ell^2(\mathbb{Z})$ be the set of all two-sided sequences $(a_i)$ in $\mathbb{C}$, such that $\sum_{n\in \mathbb{Z}} |a_n|^2 \lt \infty$. What considerations do I have to take into account when ...
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1answer
26 views

Transforming two dimensional into one dimensional binary summation

Consider two sets of real numbers $(v_i)_{i=1}^n$ and $(u_i)_{i=1}^n$. I want to find a sequence of real numbers $(t_i)_{i=1}^M$ such that $$ \sum_{s\in\left\{0,1\right\}^n}\exp\left(\sum_{i=1}^n\...
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50 views

Are there other repeating cycles of terms in this hailstone-like sequence?

I have been experimenting with a hailstone-like sequence where if a term is a multiple of 2, the next term is one half the previous term and if a term is not a multiple of two, the next term is the ...
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25 views

Better understanding of a series

Let $B=\{x\in\mathbb{R}^N:\ |x|<1\}$ with $N\geq 2$ and $x_n$ a countable dense set in $B$. Consider the function $$u(x)=\sum_{i=1}^\infty\frac{1}{2^{i}|x-x_i|^{1/2}},\ \forall\ x\in B$$ By using, ...
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1answer
34 views

As resolved the identity of this series?

Use the identity $\cos((k-\frac{1}{2})x) - \cos((k+\frac{1}{2})x) = 2\sin kx \sin \frac{x}{2}$ to show that $S_n:=\sum_{k=1}^{n}\sin kx=\frac{1}{2\sin \frac{x}{2}}(\cos\frac{x}{2}-\cos((nx+\frac{x}{...
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78 views

Reformulate summations of undefined length

I have the following equation that is not "writable" as its length depends on the variable t. So, I'd like to reformulate it. $N(t)=\sum_{i=1}^{t}\sum_{j=1}^{i}\...
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25 views

Reformulate some summations

Is there a way to reformulate the following equations (leaving N(...) alone on the left-hand side): $$N(t)=\sum_{a=1}^{t}\sum_{b=1}^{a}\sum_{c=1}^{b}\sum_{d=1}^{c}...
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2answers
553 views

A basic question on subsequential limits

Suppose we have a sequence $\{x_n\}$. Consider the set $S$ of subsequential limits of $\{x_n\}$. Suppose, for any given $\epsilon$ I do the following experiment : For each subsequential limit $x^{(k)}$...