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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.

0
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1answer
13 views

Calculating $\displaystyle{\lim_{n\to\infty}}\left(\frac{\sin(2\sqrt 1)}{n\sqrt 1\cos\sqrt 1} + …+\frac{\sin(2\sqrt n)}{n\sqrt n\cos\sqrt n}\right)$

Using the trigonometric identity of $\sin 2\alpha = 2\sin \alpha \cos \alpha$, I rewrote the expression to: $$\displaystyle{\lim_{n\to\infty}}\left(\frac{\sin(2\sqrt 1)}{n\sqrt 1\cos\sqrt 1} + ...+\...
1
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0answers
16 views

Evaluation of a sum by means of Poisson sum formula and digamma function

I have the following series: $$\sum_{n=-\infty}^{\infty}\frac{1}{(2n+1)^2\pi^2+a^2}=\frac{1}{2a}\tanh\left(\frac{a}{2}\right)$$ and on the text it is written that it can be proven by means of either ...
2
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4answers
62 views

Calculating $\displaystyle {\lim _{n\to\infty}}\frac{1+2+3+…+n-1}{n^2}$

My first attempt was using limit arithmetic, but it fails because one of the operands is infinite, so that didn't work. I then tried using the squeeze theorem: $$b_n = \frac{1+2+3+...+n-1}{n^2}$$ $$...
0
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2answers
16 views

Let $\{a_n\}_n$ be a sequence and suppose $\{a_n\}_n$ isn't bounded above. Prove that there's a subsequence $\{a_{n_k} \}_k$ such that $a_{n_k} → ∞.$

I just wanted to see if my proof worked or not, or if there was any way to improve it. Proof: In the case that $\{a_n\}_n$ is bounded below, we have $a_n\to\infty$ and so for any subsequence $\{a_{...
0
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0answers
11 views

Expression for finding common elements in two series

I have a series of the form $S_1=x(x+1)$ and another series $S_1/k$, for any $k \in \mathbb{N}$. Now I want to find the values where the elements of two series are equal. For example, let $k$ be 3, ...
0
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1answer
34 views

Find x in infinite sequence [duplicate]

We have: $$ x^{x^{x^{ x^{x ^{x ^{\dots}}}}}} = 2.$$ I tried a reasoning by recursion: For $n=1$: \begin{align} x^x &= 2 \\ \implies x\ln x &= \ln 2 \end{align} For $n=2$: \begin{align} x^...
1
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0answers
17 views

Discrete Sequence in Complex space

Let $U \subset \mathbb{C}$ an open subset. I want to construct a sequence $(a_i)_{i \in \mathbb{N}}$ contained in $U$ with following two properties: for every rational point $q=q_R+q_I i \in U$ (...
0
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1answer
21 views

Questions on a recursive sequence

A sequence is defined recursively by $b_1=−1, b_{n+1}=((4n−5)/(2n−3))b_n$. i)State if it is monotonic ii)State if it is bounded iii)Find its limit For i), $(b_{n+1}/b_n)>=1$ gives $(2n-2)/2n-...
1
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2answers
51 views

Proving $\lim _{n\to\infty}\frac{1}{ a_n} \neq \alpha$ given $\lim _{n\to\infty}a_n = 0$ and…

Given: $$\displaystyle {\lim _{n\to\infty}}a_n = 0\\\alpha \in \mathbb{R}\\a_n \neq 0$$ I'm trying to show: $$\exists \mathcal{E} > 0| \exists N \in \mathbb{N}|\forall n > N:$$ $$\left| \frac{1}{...
1
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1answer
44 views

Does the series $\sum_{n=1}^{\infty} \frac{\sin{6n}}{1+2^n}$ converge or diverge?

I'm having trouble identifying which test to use since the terms in the series oscillate between positive and negative values.
0
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1answer
21 views

In what bases is $101$ the only prime in the sequence $1,101,10101,\ldots$?

$101$ is the only prime in the sequence $1,101,10101,\ldots$ as shown in this Putnam question. I also know from studying the Collatz conjecture that $101_2$ is also the only prime in the same ...
1
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1answer
48 views

Any convergent sequence is bounded. Don’t we need to use the absolute value in this proof?

We have the following elementary result on real sequences. Any convergent sequence is bounded. This is basically the proof given in my notes: Suppose that $a_n \to a \in \mathbb{R}$. Now choose $\...
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0answers
11 views

How to find quantity of cyclical sequences of ordered pairs with permutation between adjacent pairs defined

I have manually identified possible combinations for a small set I have been working with and I'm not sure if I have overlooked some valid options. I would like to be able to calculate the quantity of ...
0
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2answers
21 views

About $l^1$ norm

By duality and Hahn Banach theorem, we know that for $x\in \ell^1$, its norm can be computed as $$\|x\|_1=\sup_{\|\beta\|_\infty=1} \left|\sum_k x_k \beta_k\right|.$$ To obtain the norm, in that ...
0
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2answers
24 views

Am I calculating my partial sums correctly? Taylor series.

So I am trying to find the MacLaurin series of $xe^{-x}$ and since I know that $$ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$ so $e^{-x} = $ $$e^{-x} = \sum_{n=0}^{\infty} \frac{(-x)^n}{n!}$$ so $...
1
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2answers
46 views

Is the sum of the series $\sum 1/3^n$ equal to $1/2$ or $3/2$?

The formula for a geometric series as I know it is $\sum ar^{n-1}$, where $r$ is the common ratio and $a$ is the first number the common ratio is multiplied with. If we're to conform to that formula,...
3
votes
2answers
69 views

Does a sequence $a_n$ converge if $|a_n-\frac{1}{n} \sum_{i=1}^n a_i| \to 0$, as $n \to \infty$?

Let $\{a_n\}$ be a real sequence. If $\lim\limits_{n\to \infty} \left|a_n - \frac{1}{n}\sum\limits_{i=1}^n a_i \right|= 0$, do we have $\lim\limits_{n\to \infty} a_n$ convergent? This is somehow an ...
0
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2answers
21 views

If you get a result other than $0$ when using the limit comparison test for series, does that tell you anything?

If $\lim_{n->\infty} a_n/b_n = c$ and $c>0$, then $a_n/b_n$ have the same convergence behavior (i.e. if one converges the other converges and if one diverges then the other diverges). But what ...
0
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2answers
74 views

Values of $\int_{0}^{1}{\frac{dt}{1+t^n}}$

May be this question has already been asked here. I’m looking for differents methods for handling this integral. Edit: I am looking for a closed form. Any suggestion or method is welcome. Initially ...
3
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0answers
26 views

How to evaluate $\sum_{n=1}^{\infty}\frac{H_{kn}^2-[\gamma+\ln(kn)]^2}{n}?$

From this post @Olivier Oloa gives the closed form for this sum $(1)$ $$\sum_{n=1}^{\infty}\frac{H_n^2-(\gamma+\ln n)^2}{n}=\frac{5}{3}\zeta(3)-\frac{2}{3}\gamma^3-2\gamma\gamma_1-\gamma_2\tag1$$ I ...
0
votes
1answer
22 views

Prove that, we can create new sequence arranging a sequence and we will still get the same limit to the new one

If the sequence $a_1, a_2, a_3, \dots$ approaches zero, prove that we can put those numbers in any order and the new sequence still approaches zero
0
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1answer
25 views

Why is $\sum_{k=1}^{6}\frac{2k-1}{36}=\frac{1}{18}\sum_{k=1}^{6}k-\frac{1}{6}$?

Why is $$ \sum_{k=1}^{6}\frac{2k-1}{36} = \frac{1}{18}\sum_{k=1}^{6}k-\frac{1}{6}? $$
0
votes
1answer
18 views

Confused about taylor series first term.

So I ama trying to find the MacLaurin series of $xe^{-x}$ and since I know that $$ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$ so $e^{-x} = $ $$e^{-x} = \sum_{n=0}^{\infty} \frac{(-x)^n}{n!}$$ so $...
0
votes
1answer
37 views

Value of $\lim_{n\rightarrow \infty}(a_{1}+a_{2}+\cdots +a_{n})$

If $\displaystyle a_{n}=\bigg(\frac{n!}{1\cdot 3 \cdot 5 \cdot 7\cdot\cdot (2n+1)}\bigg)^2.$ Then $\displaystyle \lim_{n\rightarrow \infty}\bigg(a_{1}+a_{2}+\cdots +a_{n}\bigg)$ is My Try: $$a_{...
0
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0answers
27 views

Does there exist an integer sequence that satisfies the following properties?

Does there exist an integer sequence $\{a_n\}_{n = 1}^\infty$ that satisfies the following properties: $\forall t > 1, n^t = o(a_n)$ $\forall p > 1, q > 0, a_n = o(p^{n^q})$ ? The only ...
0
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0answers
42 views

Closed form for $\sum_{j=t}^{2t}\sum_{k=0}^{t}{2t \choose t}{t\choose k}{j \choose k}{k \choose j-t}\cdots$

We manage to figure the closed form for this sum $(1)$ $t\ge2$ $$\sum_{j=t}^{2t}\sum_{k=0}^{t}{2t \choose t}{t\choose k}{j \choose k}{k \choose j-t}\frac{j^2(-1)^j}{(t+1)(2k-1)(2j-2k-1)}=3\cdot2^{2t-...
6
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3answers
246 views

What is the value of $\frac11+\frac13-\frac15-\frac17+\frac19+\frac1{11}-\dots$?

The series $\sum_{k=1}^{\infty }\frac{(-1)^{k+1}}{2k-1}=\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\dots$ converges to $\frac{\pi}{4}$. Here, the sign alternates every term. The series $\sum_{k=...
0
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0answers
19 views

Simplifying a marginal likelihood function

I have the following likelihood function $$ p( z \big| x, \lambda, \sigma) = \frac{1}{\sigma^{2}} \cdot \exp \bigg( -\frac{\big( z^{2} + \lambda^{2} \cdot x^{2} \big)}{ \sigma^{2} } \bigg) \cdot I_{0}...
2
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1answer
44 views

How many solutions does equation $\int\limits_x^{x+\frac{1}{2}} \cos \left( \frac{t^2}{3} \right) dt = 0$ have on the segment [0, 3]?

The task i'm trying to solve is: How many solutions (roots) does equation have: $$\int\limits_x^{x+\frac{1}{2}} \cos \left( \frac{t^2}{3} \right) dt = 0$$ on the segment [0, 3] ? By the moment i'...
2
votes
0answers
19 views

Expression for an intersection of two series

I have two series, one of the form $S_1=u(u+1)$ and another of the form $S_2=u(13u\pm 1)$, which leads to $S_1=2,6,12,20,...$ and $S_2= ...,114,50,12,0,14,54,120,...$. Then a third series is formed ...
1
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2answers
28 views

Find the Taylor series of this polynomial. How do I formally show radius of convergence?

The question I am asked is this: Find the Taylor series for $f(x)$ centered at the given value of a. [Assume that f has a power series expansion. Do not show that $R_n(x) -> 0.$ Also find the ...
0
votes
1answer
24 views

The convergence of a bounded sequence ${x_n}$ satisfying $x_{n+1} - \epsilon_n \le x_n$, where $\sum_{n=1}^\infty \epsilon_n$ is absolutely convergent

Statement: If a bounded sequence $\{x_n\}_{n=0}^\infty$ in $\mathbb{R}$ satisfies $x_{n+1} - \epsilon_n \le x_n$ for $n \in \mathbb{N}$, where $\sum_{n=1}^\infty \epsilon_n$ is an absolute convergent ...
-1
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0answers
19 views

Determine whether sequence is convergent or divergent

I have the following sum $\sum_{i=1}^\infty \frac{(-1)^i * |a|^\frac{1}{i}}{i} $. For what values of a does this converge? Using alternating series test if a >= 1 It converges, but for a < 1, I am ...
0
votes
2answers
25 views

Sequence\Limits Proof

Let $L = \lim_{k \rightarrow \infty}\limits x_k$. If $(x_k)_{k=0}^\infty$ is increasing, then $x_k \le L$ for all $k \ge 0$ Could anybody push me in the right direction? I've stared at this one for ...
3
votes
1answer
67 views

Find $\lim_{ n \to \infty} \frac{e^{-n}}{\sqrt{n}} \sum_{k=0}^\infty \frac{\sqrt{k+ n }}{k!} (n+a)^k$

I am trying to find the following limit: \begin{align} \lim_{ n \to \infty} \frac{e^{-n}}{\sqrt{n}} \sum_{k=0}^\infty \frac{\sqrt{k+ n }}{k!} (n+a)^k \end{align} for some fixed $a>0$. Things ...
4
votes
4answers
83 views

Proving $\displaystyle{\lim _ {n\to\infty}}\frac{6n^3+5n-1}{2n^3+2n+8} = 3$

I'm trying to show that $\exists \,\varepsilon >0\mid\forall n>N\in\mathbb{N}$ such that: $$\left|\frac{6n^3+5n-1}{2n^3+2n+8}-3\right| < \varepsilon$$ Let's take $\varepsilon = 1/2$: $$\left|...
3
votes
2answers
73 views

Simplifying $\prod_{k=3}^{n-1}\cos\left(\frac{\pi}{k}\right)$

I am looking to simplify the following, without the use of capital Pi notation: $$\prod_{k=3}^{n-1}\cos\left(\frac{\pi}{k}\right)$$ Which is meant to produce the sequence: $\left[1,\ \frac{1}{2},\ \...
0
votes
3answers
53 views

Comparison Test prove $b_n \le a_n$

On my calculus II exam, my professor wanted us to determine whether the below series was convergent or divergent. $$\sum _{n=1}^{\infty }\:\frac{1}{\sqrt[4]{n^3+1}}$$ I realized that it was most ...
0
votes
1answer
21 views

find the formula for the general nth term of the sequence.

Consider a Fibonacci type of sequence $$a_0=1~, \qquad a_1=2~, \qquad 3a_{n+1}=a_n+2a_{n-1}~, \quad n=1,2, \ldots$$ Find the formula for the general nth term of the sequence. I'm having trouble ...
0
votes
2answers
54 views

Can I find a power series representation for $\frac {1}{(1+x)^2}$ ONLY by differentiating $\frac {1}{1+x}$?

There are probably much easier to find the power series for $\frac {1}{(1+x)^2}$ than by differentiating $\frac {1}{1/1+x}$, but I think it should be possible. I've gotten extremely close, but my ...
2
votes
0answers
28 views

Looking for scientific papers about lotteries

Do you know any paper focusing on the statistical science and math behind lottery and number guessing games? I am creating a new one, where the winning numbers are known in advance, and the gain is ...
0
votes
0answers
32 views

uniform convergence on interior

I am reading Gamelin's Complex Analysis book, and stumbled upon a statement while working on a question. the question asks to show that $\sum \frac{z^n}{n}$ is not uniformly convergent for $|z| < ...
0
votes
1answer
46 views

$(3+2n)x_n = 2n x_{n-1}$

Is it possible to obtain a closed form expression of $x_n$ defined by $x_0=2/3$ and $(3+2n)x_n = 2nx_{n-1} $ for all $n\geq 1$ ?
0
votes
1answer
14 views

Can a variant of the Dirichlet eta function converge for negative numbers?

It is known that the Dirichelt eta function, defined by $$ \eta(s)=\sum\limits_{n=1}^\infty\frac{(-1)^{n+1}}{n^s} $$ converges conditionally on the open half-plane $\Re(s)>0$. This fact inspires ...
-1
votes
1answer
61 views

$\sum_{n=1}^\infty \frac{n}{4n^4+1}$ converges to?

$$\sum_{n=1}^\infty \frac{n}{4n^4+1}$$ my attempt : assumed the series is a telescopic and tried finding $t_n - t_{n-1}$ but then realized it is not a telescopic series. $$$$ //answer is given to be 0....
1
vote
1answer
33 views

Could the series $\sum (x-3)^n/n$ be seen as a power series if we consider $1/n$ as $c_n$?

I'm just trying to be sure I understand power series correctly. Would the series $\sum \frac{(x-3)^n}{n}$ be seen as a power series if we consider $\frac 1n$ as $c_n$, seeing as (taking $a$ here to be ...
1
vote
2answers
37 views

Is it correct to say that all geometric series are power series centered at $0$?

My thinking is that all geometric series are power series, since the form of a term of a power series is $c_n(x-a)^n$. If you look at $c_n$ as the starting constant and $x$ as the common ratio (and $a$...
-1
votes
1answer
42 views

Convergence of $\sum_{n=1}^{\infty} \frac{a^{n+k} n !}{n^{n+k}}$ [on hold]

Does the series $$\sum_{n=1}^{\infty} \frac{a^{n+k} n !}{n^{n+k}}$$ with $a>0$ and $k \in \mathbb{N}$ converges? Using the ratio test already found that it converges for ae, but couldn't find ...
2
votes
1answer
46 views

Fixed-point iterations for quadratic function $x\mapsto x^2-2$

Let $f(x)$ be $x^2-x-2$. I want to find the root using FPI in an interval where it will converge. I have chosen $g(x)=x^2-2$ and so $g'(x)=2x$. The convergence condition, $|g'(x)|<1$ is ...
1
vote
3answers
57 views

Determine $N$ of series $\sum_{n=1}^{N}\frac{n^n}{(2n+1)!}$ so that it differs from the actual sum by less than $\frac{1}{200}$

I can establish that: $$\frac{n^n}{(2n+1)!}=\frac{n^n}{(2n)!(2n+1)}\le\frac{n^n}{n!}$$ But $$\sum_{n=1}^{+\infty}\frac{n^n}{n!}$$ diverges (by the ratio test). And even if it converged I wouldn't have ...