Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.

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1 answer
24 views

Deriving the formula of a sequence

Let $(a,b)\in\mathbb{C^2}$ such that $(a,b)\neq0$ Let $x\in\mathbb{C/\{0\}}$ such that $x^2=ax+b$ Now, $$x^2=ax+b$$ $$\iff x^3= ax^2+bx$$ $$\iff x^3 = a(ax+b)+bx$$ $$\iff x^3 = (a^2+b)x+ab$$ $$\iff x^...
2 votes
1 answer
59 views

convergence of $\sum_n\left(1+\log\left(1+\frac1n\right)\right)^{n^2}x^n$

I used the Cauchy Hadamard theorem to get the radius of convergence finding $r=\frac1e$. Now I have to check the convergence in $|x|=r$ but I didn't manage to prove the convergence or the divergence ...
1 vote
2 answers
37 views

$\sum a_n^2 <\infty$ implies $\sum a_n^3<\infty$?

Problem : Given sequence $a_n$ s. t. $\sum_1^\infty (a_n)^2<\infty$. Then $\sum_1^\infty(a_n)^3<\infty$ ? Since $\sum a_n^2$ converge, $a_n \to 0$ where $n\to\infty$. I tried to use two ...
3 votes
2 answers
307 views

Why is 4 rare in the Van Eck sequence?

The Van Eck sequence is defined here: https://oeis.org/A181391 By far the most frequent integer at the beginning of the sequence (up to first 10 million terms) is zero. Intuitively, one would ...
0 votes
0 answers
21 views

How to find a subsequence of $x_{n+1}=1-2x_n$ with $x_1=0$ and prove it is monotone

I have a sequence $x_{n+1}=1-2x_n$ with $x_1=0$. It clearly has two subsequences which are both monotone, $(x_{n_k})$ with $n_k=2k$ and the same for $n_k=2k+1$. I tried induction to prove that the ...
0 votes
0 answers
27 views

Show that no sequence in $X\setminus\{(0,0)\}$ converges to $(0,0)$

I have been reading general topology by John L. Kelley and came across this problem in chapter 2. E. Let $X$ be the set of all non-negative integer pairs with the topology described as follows: For ...
1 vote
3 answers
48 views

Maclaurin series of $\sin(5x^2)$

I was requested to find the Maclaurin series of $\sin(5x^2)$. I attempted to find the derivatives of this function in hopes of finding a pattern. However, the derivatives become more and more ...
0 votes
0 answers
9 views

Prior distribution hyper-geometric

I started to study Decision theory and then I tried to do the exercises in the book Optimal Statistics Decision, Morris H. DeGroot, but I don't know how to solve the following question: Question: ...
0 votes
0 answers
16 views

splitting sequence into two

From exercise $10.4.34$ in Tom Apostol's Calculus vol $1$ book, we know that: $\tag{1} \lim_{N \rightarrow \infty} \sum_{k = 1}^N \frac{1}{2N} \sin \frac{k \pi}{2N} = \int_0^{\frac{1}{2}}\sin(x \pi) ...
1 vote
5 answers
57 views

Finding general formula for the sequence $b_{n+1} = 2\cdot b_n + 2^{n-1}$

In an algorithm I'm developing for one of my computer science classes, I need to find the general formula for each term of the sequence $T_n$. This sequence is defined in the following way: Define ...
-1 votes
0 answers
24 views

Is chaos possible for a real sequence that converges to a finite number of cyclic points?

Let us suppose that we have a sequence $a_n$ over the real numbers. This sequence will reach a periodic cycle with a period $r>0$. Such a cycle is described as follows: $$\lim_{n \to \infty} a_{i+n*...
1 vote
1 answer
28 views

Are there bounded monotone sequences whose product is not monotone?

I was wondering whether or not the product of two monotone and bounded sequences must be monotone. I can think of e.g. $(x_n)=(-1,-1,1,1,1,…)$ and $(y_n)=(1,0,-1,-1,-1,…)$ that are clearly both ...
2 votes
0 answers
38 views

Prove $\sum_{n=-\infty}^\infty (-1)^n \frac{1-2z}{(1-2z)^2-4n^2}=2\sum_{n=0}^\infty (-1)^n \frac{1+2n}{(1+2n)^2-4z^2}$ [duplicate]

I should prove that $$\frac{\pi}{2}\sec\pi z=2\sum_{n=0}^\infty (-1)^n \frac{1+2n}{(1+2n)^2-4z^2}.$$ I am given the following hint: Replace $z$ by $\frac{1}{2}-z$ in $$\pi\csc\pi z=\frac{1}{z}+\sum_{...
3 votes
1 answer
43 views

Is $x_{j} \bmod v$ dense in $[0,v]$?

Suppose that $(x_{j})_{j \in \mathbb{N}}$ is an unbounded sequence inside $\mathbb{R}$. Does there exist a $v>0$ so that $x_{j} \bmod v$ is dense in $[0,v]$? Note here that $a \bmod b := a- \lfloor ...
3 votes
1 answer
22 views

Induction problem involving two base cases and two assumptions of truth in recurrence relation

I am unsure why I have to use two base cases of n=1 and n=2 and also assume truth for n=k and n=k+2 to prove by induction that n=k+2 is true. I was trying to solve this question below: A sequence is ...
0 votes
1 answer
30 views

Comparing sums of fractions.

Consider a number n. Out of all the fractions arising from the scenario by reducing some x from n and adding the ratio of the removed to the current n till n becomes 1, why does the sum be maximum if ...
0 votes
0 answers
20 views

Product of a bounded sequence by a null sequence is a null sequence

Suppose $\{a_n\}$ is a bounded sequence and $\{b_n\}$ is a sequence converging to zero. Show that $\{a_nb_n\}$ converges to zero. My attempt: $\textit{Proof:}$ Suppose $\{a_n\}$ is a bounded sequence ...
2 votes
2 answers
144 views

Where is the flaw in my approach? : AMC12A 2010 Problem 20

On the problem below, I am unsure why my answer was wrong. Arithmetic sequences $\left(a_n\right)$ and $\left(b_n\right)$ have integer terms with $a_1=b_1=1<a_2 \le b_2$ and $a_n b_n = 2010$ for ...
-3 votes
1 answer
23 views

How can I find the next sum? [closed]

I've just read about the behaviour of infinite series, but it doesn't seem to work for this problem: 1+5+9+...+(4n-3)=
0 votes
0 answers
23 views

Limit of $z_n = n(1-\cos(\theta/n) - i \sin(\theta/n))$

I wish to find the limit of the sequence $z_n = n(1-\cos(\theta/n) - i \sin(\theta/n))$, for some fixed $\theta$. To do so, I have separated the real and imaginary parts, observing Real Part: $\text{...
0 votes
0 answers
31 views

Proof that if $x_n$ tends to infinity, then $y_k=x_k^2-2x_k$ tends to infinity

Proof that if $x_n$ tends to infinity, then $y_k=x_k^2-2x_k$ tends to infinity I know that since $x_n$ tends to infinity, for all M there exists an N such that $x_n>M$ for all $n>N$. However, ...
1 vote
3 answers
54 views

How to prove that the sequence $y_{n+1}=1-2y_n$ where $y_1=0$ has no convergent subsequences

How to prove that the sequence $y_{n+1}=1-2y_n$ where $y_1=0$ has no convergent subsequences? I don't know where to even begin with this proof. I'm assuming I need to prove that $|y_n|$ converges to ...
0 votes
0 answers
25 views

Proof explanation, if $a_n \leq c \text{ } \forall n \in \mathbb{N}$ for $c \in \mathbb{R}$, then $a \leq c$ also holds

Let $(a_n)_{n \in \mathbb{N}}$ be a convergent sequence with $\lim_{n \to \infty} a_n = a$. Prove, if $a_n \leq c \text{ } \forall n \in \mathbb{N}$ for $c \in \mathbb{R}$, then $a \leq c$ also holds....
-1 votes
1 answer
23 views

How to prove that c(n) = -(c(n-1))? (Fibonacci)

$a(n) = a(n−1)+a(n−2)$ with $a(0) = 1, a(1) = 1$ $c(n) = a(n)a(n−3) − a(n−1)a(n−2)$ How can I prove that c(n) equals -c(n-1)? And that c(n) = (-1)^(n+1)?
5 votes
4 answers
3k views

why $N>\frac{1}{\epsilon}$?

i am asking here the most simple and dumbest question ever. once i was reading about convergence yesterday night, i came to the notion which beats me over times: The archimedean axiom the text was ...
4 votes
1 answer
118 views
+100

Is $\|\cdot\|_{\pi} = \|\cdot\|$?

Let $\ell^{1}(\mathbb{N})$ be the space of complex-valued sequences such as: $$\|a\|_{\ell^{1}} = \sum_{n\in \mathbb{N}}|a_{n}| < +\infty$$ and set $\ell^{1}(\mathbb{R})\otimes \ell^{1}(\mathbb{R})$...
1 vote
2 answers
69 views

Prove if sequence $s_{2n}$ converges, then also $s_n$ converges

I'm having some trouble understanding why it is justified to introduce substitution when computing the limits of sequences. The definition of sequence conversion is a usual, where $m$ are positive ...
0 votes
0 answers
26 views

Converging Sequence Quintic

Let $a_{n+1} = \frac{3n^5 + 20n^3 + 7}{2n^5 - 1}$, $l = \frac{3}{2}$. Prove that $a_n$ converges to $l$. I used the absolute value of $a_{n} - l$, to obtain $\frac{40n^3 + 17}{4n^5-2}$. I then ...
0 votes
0 answers
6 views

Z Transform - Radii of convergence of different series expansions of a rational function

This question relates to inversion of the Z transform as described by Gabel & Roberts' Signals and Linear Systems 3rd edition, pg 204. The first method considered is Series Expansion. For the ...
2 votes
2 answers
162 views
+50

prove that $\lim\limits_{n\to\infty} x_n = \lim\limits_{n\to\infty} y_n$

For $x,y>0$, define two sequences $(x_n)$ and $(y_n)$ by $x_1=x,y_1=y$ and $x_{n+1}=(x_n+y_n)/2$ and $y_{n+1}=\sqrt{x_ny_n}$. Prove that $\lim\limits_{n\to\infty} x_n = \lim\limits_{n\to\infty} y_n=...
0 votes
0 answers
49 views

How can I calculate this repeating power?

I'm considering the stochastic tree. When root node is activated, the $n$ child nodes get the signal. But the probability of activation is $p$. $d$ is index of layer and starts from 1 that is ...
0 votes
1 answer
31 views

Arithmetic Mean

Based on the definition of the arithmetic mean. AM between two non-consecutive terms $x_1$ and $x_2$ of the AP sequence is the following $$\frac{x_1 + x_2}{2}$$ But I'm a little confused on how this ...
2 votes
2 answers
91 views

Neccesary condition for an integral to be finite

Suppose that $(X,\Sigma,\mu)$ is a measurable space and $f$ a non-negative measurable function such that $$ \int_{X}{f}< +\infty $$ I want to prove that $\sum_{n=0}^{\infty}{2^n \mu( \left\{{x:f(x) ...
1 vote
1 answer
50 views

Substitution $t=-t$ in series

Let's say I want to make a substitution $t=-t$ in a series $$\sum_{t=-\infty}^{\infty}a_t$$ The result of this substitution should be $$\sum_{t=\infty}^{t=-\infty}a_{-t}$$ correct? Based on the ...
3 votes
1 answer
69 views

What's The Limit of The Recursion $a_{n+1} = \sqrt{2+\sqrt{a_n}}$ with $a_0 = \sqrt{2}$?

I'm following a study guide for an exam and the very first question is: Show that the sequence defined by $a_{n+1} = \sqrt{2+\sqrt{a_n}}$ with $a_0 = \sqrt{2}$ is increasing, bounded and calculate it'...
8 votes
4 answers
421 views

Asymptotics of the sum $\sum_{n=1}^\infty \frac{x^n}{n^n}$

How does the sum $$f(x)=\sum_{n=1}^\infty \frac{x^n}{n^n}$$ behave asymptotically as $x\to\infty$? It appears that $f(x)$ asymptotically dominates any polynomial and is dominated by any exponential, ...
1 vote
1 answer
70 views

Fractional part of $2\sqrt{b^n(b^{n+1}-1)}$ dense in $(0,1)$

I'm looking to prove that the fractional parts of the sequence $x_n = 2\sqrt{b^n(b^{n+1}-1)}$ where $n \geq 1$ are dense in $(0,1)$ for all integers $b \geq 2$. I know that each member of the sequence ...
-6 votes
1 answer
23 views

Does the sequence converge or diverge? And why? [closed]

Does the sequence $a_n = \left|(-1)^n \right|$ converge or diverge? And why? If it converges what value does it converge to?
3 votes
4 answers
165 views

Prove the sequence of three real numbers

If $a,b,c$ are non zero real numbers satisfying $$(ab+bc+ca)^3=abc(a+b+c)^3$$ then prove that $a,b,c$ are terms in $G.P$ My work: I assumed that they are in $G.P$ and so assumed $b=ak$ and $c=ak^2$ ...
1 vote
0 answers
36 views

How to find the value of the given limit with summation?

It was asked to find the value of the following limit: $$ \lim _{n \rightarrow \infty} \frac{96}{n^4}\left[1\left(\sum_{k=1}^n k\right)+2\left(\sum_{k=1}^{n-1} k\right)+3\left(\sum_{k=1}^{n-2} k\right)...
1 vote
2 answers
39 views

Comparison test for proving convergence

I'm trying to determine whether this series is convergent or divergent: $$ \sum_1^\infty \frac{e^{1/n}-1}{n} $$ I thought directly that $e^{1/n} \geq 1$ and therefore $\frac{e^{1/n}-1}{n} \geq \frac{...
6 votes
2 answers
121 views

Check if this series is convergent or not

I've been going crazy for a long time in determining if this series converges or diverges. Most likely it converges. $\sum_{n=1}^\infty (\frac{n}{2} sin\frac{1}{n})^\frac{n^2+1}{n+2}$ I am stuck in ...
7 votes
1 answer
306 views

Is there a connection between the sum of $\sum_{n=1}^\infty \frac{\ln(n+1)-\ln(n)}{n}$ and the Riemann zeta function?

It can be shown that for every integer $p\geq 0$, this integral identity holds: \begin{align} \int_1^\infty\frac{1}{x^{p+2}\lfloor x\rfloor}\text{ }dx &= -1+\frac{1}{p+1}\sum_{m=2}^{p+2}\zeta(m)\\ ...
3 votes
2 answers
189 views

How to tackle the integral $\int_{0}^{\infty} \frac{\ln x}{x^{n}-1} d x$?

In my post, I started to investigate the integral $\displaystyle \int_{0}^{\infty} \frac{\ln x}{x^{2}-1} d x$. Fortunately, $$\displaystyle \int_{0}^{\infty} \frac{\ln x}{x^{2}-1} d x =2 \int_{0}^{1} ...
2 votes
1 answer
32 views

If the quotient of limits converge to 0 then there exists $M \in \mathbb{N}$ such that for all $n \ge M$, $y_n > |x_n|$.

Suppose $\{x_n\}$ and $\{y_n\}$ are sequences (not necessarily convergent) such that $y_n > 0$ for all $n \in \mathbb{N}$ and $$\lim_{n \rightarrow \infty} \frac{x_n}{y_n} = 0$$ Prove that there ...
1 vote
2 answers
36 views

Prove $\sum_{x=1}^\infty xr^x = {r\over r-1}$ by taking the limit of the partial sum formula

There are multiple ways to derive $\sum_{x=1}^\infty xr^x = {r\over (r-1)^2}$ mentioned here How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$? but none of them show the derivation by taking the limit of ...
-1 votes
1 answer
55 views

How can I Taylor expand an expression that has a square root under exponent of -1/2?

So, in a quantum mechanics homework I have, I have an eigenstate $|\psi\rangle$ given by $$|\psi\rangle=\left(1+\frac{\Omega^2}{\left(\delta+\sqrt{1+\left(\frac{\Omega}{\delta}\right)^2}\right)^2} \...
2 votes
1 answer
81 views

Closed-form or integral representation for $\sum_{n=0}^\infty\frac{e^{-\lambda}\lambda^n}{n!}\phi(z-n;\mu;\sigma)$.

Let $Z=X+Y$, where $X\sim\operatorname{Poisson}(\lambda)$ and $Y\sim\mathcal N(\mu,\sigma^2)$ are independent. The density of $Z$ can thus be expressed as an infinite component Gaussian mixture of ...
0 votes
1 answer
30 views

Finding the cluster points of a sequence in $\mathbb{R}^3$

I haven't found many examples of how to find cluster points for sequences of real numbers instead of sets and for some reason I've found more confusing to do this with sequences than with sets. ...
5 votes
2 answers
57 views

Limit with Arithmetico-geometric sequence

It is given that $$ L=\lim _{k \rightarrow \infty}\left\{\frac{e^{\frac{1}{k}}+2 e^{\frac{2}{k}}+3 e^{\frac{3}{k}}+\cdots+k e^{\frac{k}{k}}}{k^2}\right\} $$ I tried solving it but I am stuck on this, ...

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