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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Find the limit of $4^n a_n$, for the recurrent sequence $a_{n+1}=\frac{1-\sqrt{1-a_n}}{1+\sqrt{1+a_n}}$

Given the recurrence relation $$ a_{n+1}=\frac{1-\sqrt{1-a_n}}{1+\sqrt{1+a_n}} $$ which is easy to find $$ a_n\to0, \quad b_n=\frac{a_{n+1}}{a_n}\to\frac1{4} $$ hence $a_n\sim4^{-n}$, or with some ...
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Find $\alpha$ and $\beta$ such that $\sum_{n=1}^{\infty}\arctan({\frac{1}{n^{\alpha}}})-e^{\frac{2}{n^{\beta}}}+1$ converges

I want to find $\alpha\in\mathbb{R}$ and $\beta\in\mathbb{R}$ such that the following series converges: $$\sum_{n=1}^{\infty}\arctan({\frac{1}{n^{\alpha}}})-e^{\frac{2}{n^{\beta}}}+1$$ I have thought ...
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-3 votes
0 answers
14 views

$\{x_n\},\{y_n\}$ two monotone sequences $\in \mathbb R$ such that $\sum x_n y_n$ converges. Which of the following is/are true? [closed]

a. At least one of $\{x_n\},\{y_n\}$ is bounded. b. $\{x_n\},\{y_n\}$ both are bounded. c. At least one of $\sum x_n,\sum y_n$ is convergent. d. $\sum x_n,\sum y_n$ both are convergent.
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1 answer
37 views

How is this series diverging given this approxiamation?

I am given that a series follows the following formula: $$\sum_{n=1}^{\infty} 1/\sqrt{(n^2+n)} %$$ I approxiamate it with the following integral: $$\int_1^∞ 1/x \, dx = ln(∞)-ln(1)$$ Which simplifies ...
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0 votes
1 answer
26 views

Confusion to a solution to finding the Principal Part of a Laurent Series

Hello I am trying to find the Principal part at the pole $z = 1$ of the function $\frac{z}{(z^2-1)^2}$. Clearly, the function has a pole at $z=1,-1$, and by a theorem, $z=1$ is an order $2$ pole. Now ...
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  • 147
1 vote
1 answer
59 views

Find all $x \in \mathbb{R}$ such that $\sum_{n=2}^{\infty} \frac{\sin(nx)}{\log n}$ converges.

Find all $x \in \mathbb{R}$ such that $$\sum_{n=2}^{\infty} \frac{\sin(nx)}{\log n}$$ converges. My work: I tried to find for which $x$ partial sum $$\sum_{n=2}^{m} \sin(nx)$$ is bounded because, we ...
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  • 311
2 votes
1 answer
49 views

Convergence of random series $\sum_{k=1}^{n}\prod_{i=k}^{n} X_i$

If $X_i$ are i.i.d. random variable such that $|X_i|\leq 1$ almost surely for any $i$. Define $$S_1 = X_1$$ $$S_2 = X_2(1+S_1) = X_2 +X_1X_2$$ $$S_3 = X_3(1+S_2) = X_3 + X_2X_3 + X_1 X_2 X_3$$ $$S_3 = ...
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0 votes
1 answer
36 views

Hint how to proceed to prove convergence of the senquence

Let $1\leq x_1\leq x_2 \leq 2$ and $x_{n+2} = \sqrt {x_{n+1}x_n} , n \in N$ Show that $x_n$ converges. I got the value for $x_n$ as $x_n = \frac{\sqrt{x_1}}{\sqrt{x_{n-1}}}x_2$ , please guide how to ...
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0 votes
0 answers
22 views

$n$-th partial sum -- generating functions

Consider a finite series $S_n:=\sum_{i=0}^n i\cdot2^i$. The generating function for infinite formal series $\sum i\cdot x^i$ is $$a(x)=x\left(\frac{1}{1-x}\right)'=\frac{x}{(1-x)^2},$$ hence $$\frac{a(...
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1 vote
1 answer
43 views

Asymptotics for an exponential sum

I am trying to find an equivalent of : $\displaystyle{S_n=\sum_{k=1}^n}e^{i\sqrt k}$ I tried to elucidate the asymptotic behaviour of the subsequence : $$S_{n^2-1}=\sum_{k=1}^{n^2-1}e^{i\sqrt k}=\sum_{...
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  • 7,079
0 votes
2 answers
32 views

Prove that for the sequence $a_n=2a_{n-1}, \forall n\geq 2 \iff a_n=\sum_{i=1}^{i=n-1}(a_{i})+1$ by induction

Prove that for the sequence $a_n=2a_{n-1}, \forall n\geq 2$ with $a_1=1$ $\iff a_n=\sum_{i=1}^{i=n-1}(a_{i})+1$ I thought about a proof by induction. Assume that the hypothesis that $a_n=2a_{n-1}$ ...
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  • 389
-1 votes
0 answers
18 views

sum of geometric sequence is Sn=(3^n)+a, what's the value of a [closed]

I know the formula for the sum of the geometric sequence is a1(1-r^n)/n but don't know how to apply it here...
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  • 1
1 vote
1 answer
32 views

Finding power series expansion at a point that is not 0

I want to find the power series expansion of $\frac{z}{(1-z)^2}$ at $z=2$. I tried to separate the $z(\frac{1}{(1-z)^2})$ and $\frac{1}{(1-z)^2}$ is just the geometric series of $\sum kz^{k-1}$ but ...
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  • 147
-1 votes
1 answer
11 views

Solving a non-homogeneous recurrence sequence [closed]

enter image description here please solve this i am getting alwaya an no plus in my iteration
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0 votes
0 answers
41 views

How to prove that $ \sum_{n=1}^{\infty} \frac{(-1)^{[\log n]}}{n}$ diverges? [duplicate]

How to prove that $\sum_{n=1}^{\infty} \frac{(-1)^{[\log n]}}{n}$ diverges? I've tried something with the following $$\log n - 1 <[\log n] \leq \log n$$ but I haven't got anything from that. Can ...
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  • 311
0 votes
1 answer
47 views

Problem on studying $\sum_{n\geq 1}\frac{n^{\log{n}}}{\sqrt{n!}}\frac{\tan{n}}{|\tan{n}|+n}$, comparison criterion

To study the convergence of the series $\sum_{n\geq 1}\frac{n^{\log{n}}}{\sqrt{n!}}\frac{\tan{n}}{|\tan{n}|+n}$ I have thought that: $$\frac{n^{\log{n}}}{\sqrt{n!}}\geq \frac{1}{\sqrt{n!}}\,\, \forall ...
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  • 999
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0 answers
20 views

Is the limit of this sequence Lipschitz-continuous?

Suppose I have sequences $(f_{i,n})_n$ for $i=0,1,..,m$ of $M$-Lipschitz functions from $\mathbb{R}$ to $\mathbb{R}$. Each of those uniformly converges to an $M$-Lipschitz function $f_i$. Now ...
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1 vote
0 answers
31 views

How do I prove that this sequence is increasing?

If $a_k = \sum_{i=0}^{k-1} \left( \frac{1}{a_k} \right)^i$ for $a_k \neq 0$, how do I show that $a_k\in\mathbb{R}$ (strictly) increases as $k\in\mathbb{N}$ increases? I've tried induction but I just ...
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1 vote
1 answer
36 views

Is there $\sigma_\infty$ such that $\sigma_\infty$ is a subsequence of $\sigma_n$ for all $n \in \mathbb N$?

I have come across this "Diagonal method" in this lecture note. Theorem: Let $S$ be a non-empty set. For each $n\in \mathbb N$, let $\sigma_n$ be a sequence of elements of $S$. We denote by ...
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1 vote
0 answers
22 views

Convergence Rate of the tail of a series.

Consider a sequence $(a_k)_{k \in \mathbb{N}}$. Then the (absolute) convergence of the series $\displaystyle\sum_{k=1}^\infty a_k$ implies that that the sequence of tail series has limit zero, i.e. $$\...
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0 votes
0 answers
44 views

Show that $\varlimsup\limits_{n\rightarrow \infty}(\frac{1+a_{n+1}}{a_n})^{n} \geq e$ for any positive sequence $(a_n)$ [closed]

"Show that $\varlimsup\limits_{n\rightarrow \infty}(\frac{1+a_{n+1}}{a_n})^{n} \geq e$ for any positive sequence $(a_n)$" I found this problem on page 148 of Zorich's Mathematical Analysis (...
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1 vote
0 answers
23 views

Prove uniform convergence of bounded sequence

Supose we have a sequence of real continuous functions on an interval $[0,b] \in \mathbb{R}$ $(f_n)_{n=0}^{\infty}$ (hence uniformly continuous). Moreover the sequence has the following properties: ...
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0 answers
50 views

Proof of equality given by wolfram alpha

By 'playing' with wolfram alpha and entering this sum $$\sum_{n=0}^{+\infty}(-1)^n\left(\frac{1}{1+nx}+\frac{1}{x-1+nx}\right)$$ I found this alternative form that interest me: (since I don't know the ...
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0 votes
0 answers
32 views

Total boundedness and Cauchy sequences.

I am reading an article about how every totally bounded set has a Cauchy subsequence contained in it. I've included a screenshot. I understand every part except Why is $d(q_{n}^{(k)}, q_{n'}^{(k)}) &...
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0 votes
1 answer
24 views

Assumption of continuously differentiable function in the Lyapunov Stability Criterion

According to the proof of Lyapnuov's theorem given in [1] the assumption of continuity of partial derivatives is necessary to prove asymptotic stability while for simple stability it is not. I wonder ...
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0 votes
0 answers
22 views

How to prove cosine law using the power series expansion of cosine and sine?

How to prove the equation $\cos(x + y) = \cos(x) \cos(y) -\sin(x) \sin(y)$ using the power series expansions \begin{equation*} \cos(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!}, \qquad \sin(x)...
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0 votes
1 answer
24 views

Quick proof regarding a question regarding the open, closed, etc. property of sets

I was hoping that you guys can confirm that my rough working is correct over the following question: Let $S$ be the set defined as $$ S = \bigcup_{n=1}^{\infty} \biggl( \frac1{n+1}, \frac1n \biggr) $$ ...
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1 vote
0 answers
28 views

Exchanging sum and integral in the computation of the completed zeta function

In analytic number theory, when deriving the integral formula for the completed zeta function, one makes use of the following line of equation: $$\pi^{-s}\Gamma(s)\zeta(2s)=\sum_{n=1}^{\infty}\int_0^{\...
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  • 2,122
-4 votes
1 answer
34 views

How the converging geometric series discovered or created? [closed]

It is easy to create new diverging geometric series. But how the converging geometric series evolve? Do they already exist in nature and waiting to be discovered or can we create them artificially?
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  • 173
1 vote
1 answer
69 views

Sum of n + n(n-1) + n(n-1)(n-2) + ... + n!

This is to work out the time complexity of a computer science problem (write an algorithm to calculate the permutations of an array of n distinct integers). Various answers on leetcode say the sum ...
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1 vote
0 answers
27 views

Find the difference equation given the general solution $y(k) = c_{1}5^{k} + c_{2}(-5)^{k} + c_{3}6^{k}$

Given that $y(k) = c_{1}5^{k} + c_{2}(-5)^{k} + c_{3}6^{k}$, is the general solution to a difference equation, how do you work backwards to find the difference equation?
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0 votes
0 answers
63 views

How to compute $\displaystyle\prod_{i = 1}^{\infty} \frac{x^{i}}{e^{i!}}$?

In my text book it is stated (without any explanation) that $$ \prod_{i = 1}^{\infty} \frac{x^{i}}{e^{i!}} = e^{e^x - 1} $$ and I can't really think of how one can show this.
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  • 33
0 votes
0 answers
38 views

Find the Taylor series at $a=\frac{\pi}{2}$ for $f(x)=x\sin(x)$

I'm trying to find the taylor series at $a = \frac{\pi}{2}$ for $f(x)=x\sin(x)$ The problem is that I don't know if my answer is right.. Could somebody check/correct this $f(x) = x\sin(x)$ let $x = t+\...
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4 votes
1 answer
146 views
+50

New formulae for the Riemann Zeta Series

Back in late April I arrived at the following formula for when $x+b < -1$: $$\tag{1}\boxed{ \sum_{a=1}^{\infty}\sum_{k=0}^{\infty}\frac{(k+a)^{x+b}}{a} = \sum_{k=0}^{\infty}\sum_{g=0}^{k}\frac{b^{...
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0 votes
0 answers
22 views

double series convergence with conditions [closed]

Let $\sum ^{\infty }_{n=1}a_{n}$ be a positive convergent series. for every $(i,j) \in\mathbb{N}\times\mathbb{N}$ we define $c_{i,j}=\begin{cases}\dfrac{2^{i-1}a_{i}}{2^{j}}\ \ \ \ j\geq 1\\ \ \ \ \ 0\...
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0 votes
1 answer
29 views

Polynomial approximation for small x

I found this approximation made in a paper. The equation was essentially, $H = \sum_{i=0}^{N-1}\frac{\lambda}{2}((x^2+a^2)^{1/2}-a)^2$ and that for small x this could be approximated as $H = \sum_{i=0}...
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5 votes
1 answer
95 views

If Series $\sum \sqrt\frac{a_n}{n}$ converges then $\sum a_n$ converges?

Given $a_n>0$. If Series $\sum \sqrt\frac{a_n}{n}$ converges then $\sum a_n$ converges? My approach: Ihad used atmost every method like 1)AM and GM inequality 2) cauchy Swartz inequality 3) by ...
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  • 61
0 votes
0 answers
27 views

Aproaches to solving series problems

So one of my homework questions is to find for which values of $\beta,\theta\in\mathbb{R} $ Does the series ${\sum}\frac{\cos\left(n\theta\right)}{n^{\beta}}$ converge. My approach was to look at ...
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  • 1
1 vote
0 answers
31 views

Can we use comparison test for testing the convergence of following series $\sum_{n=1}^\infty \sin(\pi \sqrt{n^2+k^2}) , k\in \mathbb{Z}$?

I am not sure if I've understood this well, problem is following: Test the convergence of following series $$\sum_{n=1}^\infty \sin(\pi \sqrt{n^2+k^2}) , k\in \mathbb{Z}$$ My question is how can we ...
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  • 311
1 vote
1 answer
30 views

Maclaurin series, find the tenth derivative

The problem is as follows: Find the Maclaurin series of $$\begin{cases} \frac{\sin(x)}{x},& x \neq 0 \\ 1,& x=0 \end{cases}$$ and then find $f^{10}(0)$. I figured out the series, if $x\neq 0$ ...
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  • 21
-1 votes
0 answers
19 views

How many different X arrays are there without Y consecutive Z's [closed]

You are given a positive integer $N$, $k$, $l$ and distinct symbols $x_1, x_2, x_3 ... x_k$. $f(N)$ indicates the number of arrays (consisting of $x_1, x_2, x_3 ..., x_k$) of length $N$ that don't ...
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  • 9
1 vote
0 answers
25 views

Uniform convergence on [a, b] of series of continuous functions implies that series is continuous on (0, b]

(1) Let's say we have a series $f = \sum_{n=1}^\infty f_n$ which converges uniformly on every closed interval $[a, b]$ where $a > 0$ is arbitrary and $b$ is fixed. Moreover $f_n$ is continuous for ...
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2 votes
0 answers
44 views

Convergence of $\sum_{n=1}^\infty\frac{\sin(n)}{n}$ [duplicate]

I need to prove the convergence of: $\sum_{n=1}^\infty\frac{\sin(n)}{n}$. To do it I wanted to use the alternating series test. I know that $a_n=1/n\overset{n\rightarrow \infty }{\rightarrow} 0$ but ...
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0 votes
1 answer
73 views

If $a_n$ is a positive sequence and $\lim \limits_{n \to \infty}a_n=0$ then there exists $N>0$ such that $(a_{N+n})$ is decreasing

If $a_n$ is a positive sequence and $\lim \limits_{n \to \infty}a_n=0$ then there exists $N>0$ such that $(a_{N+n})$ is decreasing The first way is by contradiction with an example: let $ a_n= \...
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  • 825
1 vote
1 answer
46 views

Explanation about this probabilistic proof that leads to $log_RN$

I am reading a proof based on (probabilistic) analysis on how a specific data structure performs for the case of search miss. The base assumption is that the probability that each of the $N$ keys in a ...
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  • 1,539
3 votes
1 answer
31 views

Prove $\lim \inf (a_n)+\lim \inf (b_n) \leq \lim \inf (a_n+b_n)$ using subsequence

Let $\left(a_{n}\right)_{n=1}^{\infty},\left(b_{n}\right)_{n=1}^{\infty}$ be bounded sequences. Prove that $\underset{n\rightarrow\infty}{\lim\inf}\left(a_{n}\right)+\underset{n\rightarrow\infty}{\lim\...
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5 votes
2 answers
75 views

$a_{m^2}=a_m^2,a_{m^2+k^2}=a_ma_k$ sequence

Sequence $\{a_n\},n\in\mathbb N_+$ with all terms positive integers satisfy $a_{m^2}=a_m^2,a_{m^2+k^2}=a_ma_k$. Find $\{a_n\}$. I suppose all terms of $\{a_n\}$ are $1$. This problem makes me think ...
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  • 379
1 vote
2 answers
76 views

Closed form for a sequence 2, 7, 18, 41, 88, 183, 374

How can I generalize the sequence $2, 7, 18, 41, 88, 183, 374$, so to be able to retrieve any $n$th term. Initially I thought that this sequence is quadratic, but quadratic sequence requires a ...
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  • 35
1 vote
1 answer
32 views

Do sequences with the same tail have equivalence relation and converge to the same limit?

I have a problem with the following exercise: There are two sequences, $ a_i = -11, -12, -13, -14, -15, -16, -17, -18, -19....$ $ b_i = 1000, 10001, -16, -17, -18, -19...$ And the sequence $ b_i $ ...
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2 votes
0 answers
41 views

Show that for any sequence $\{γ_k\}$ convergent to zero, there exists a sequence $\{x(γ_k)\}$ which tends to $0$

Let $x\in \mathbb{R}$. Consider the following problem $$\text{minimize}\quad x\quad\text{s.t}\quad x^2\geq 0 \quad \text{and}\quad x+1\geq0$$ The optimal solution is $x=-1$ Define the (primal) ...
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