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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.

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0answers
7 views

Proof Verification: A monotonically increasing sequence that is bounded above always has a LUB

Problem: Prove that a monotonically increasing sequence that is bounded above always has a least upper bound. This theorem is present in calculus and real analysis books along with proofs for it. I ...
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2answers
26 views

Method for finding a sequence's limit

i'm trying to find the following limit, if it exists, $$\lim_{n→ ∞} \frac{(n+7)^{n-5}}{n^n}$$. Now, I've tried division like $$\lim_{n→ ∞}\frac{|n+1|}{|n|}$$, or dividing by the highest power, but I ...
4
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3answers
40 views

Constructing an arithmetic progression so that $\sum_{i=1}^n f(x_i) =0$

Let $f: \mathbb{R} \to \mathbb{R} $ be a continuous function so that $ \exists a, b \in \mathbb{R} $ with $f(a) f(b) <0$. Prove that $\forall n>2 \exists$ an arithmetic progression $x_1<x_2&...
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1answer
39 views

Laurent series problem on $f(z)=\frac{ z }{ z^2-z-2 }$

I have problems with computing Laurent series of the function $f(z)=\frac{ z }{ z^2-z-2 }\quad$ in the ring centered in $0$ containing point $1+i$. I also have to find the radius of convergence of ...
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1answer
33 views

Is there an infinite sequence that converges only for a specific x?

Is there a known infinite sequence $\sum_{i=0}^\infty f_i(x)$ that converges only for a specific $x$, and otherwise diverges?
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0answers
14 views

$\dfrac{1}{2d}\mathbb{E}N(x_{i}) \leq N(x^\prime_{i}) \leq \mathbb{E} N(x_{i})$.

1) Consider $n$ Points, $x_{1}, x_{2},...x_{n}$ distributed uniformly in $[0, 1]^d$. Term $d$ is the dimension. 2) Then, I construced a grid points $x^\prime_{1}, x^\prime_{2},...x^\prime_{n}$ that ...
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1answer
26 views

What to do with limits of infinite sequences with trigonometric functions?

Consider the sequence $$ a_n = \left (\frac {1+\cos (n)}{2+\cos (n)}\right)^{2n-\log(n)} $$ If Cauchy's convergence criterion is used (square root of $a_n$ - the exponent approaches 2 as x approaches ...
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0answers
29 views

Rate of divergence of finite sum

What is the rate of divergence of $$f(n) = \sum_{k=1}^n \frac{1}{k^\alpha}$$ when $0 < \alpha <1$? I know that $f(n)$ diverges as fast as $ln(n)$ when $ \alpha =1$. I'm wondering whether ...
1
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1answer
33 views

Completeness of a specific Hilbert space

I am reading Zehnder's book "lectures on dynamical systems". In the chapter 7 he defines a space and states that it is a Hilbert space. I am struggling to show that the space is indeed complete. Here'...
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0answers
18 views

Prove that the limit of a continuity function equals the limit of the corresponding sequence

following question: Let $U$ be a monotonically increasing function. If $\lim\limits_{n \to \infty} \frac{U([t]x-b_{[t]})}{a_{[t]}} = D(x)$ holds for every continuity point $x$ of D with $[t]$ being ...
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1answer
23 views

Is $a_n = (-1)^{[ \log n]}$ is oscillatory sequence?

I know that $a_n =(-1)^n$ is oscillatory sequence as sequence osciallate between $-1$ and $+1 $ I have some doubt in my mind that Is $a_n = (-1)^{[ \log n]}$ is oscillatory sequence ...
2
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1answer
47 views

conversion of Binomial identity into series sum

Prove that $$\binom{n}{1}(1-x)-\frac{1}{2}\binom{n}{2}(1-x)^2+\frac{1}{3}\binom{n}{3}(1-x)^3+\cdots \cdots +(-1)^{n-1}\frac{1}{n}(1-x)^n$$ $$=(1-x)+\frac{1}{2}(1-x^2)+\frac{1}{3}(1-x^3)+\cdots +\frac{...
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1answer
26 views

Proving a limit converges to -3 using definition of convergence.

so I have the problem $\lim\limits_{n \to oo} \frac{2-3n^2}{n^2+2n+1}$. I have to prove this using the epsilon definition. So I know the limit equals -3. So I do |$\frac{2-3n^2}{n^2+2n+1}$ + 3 | < $...
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0answers
14 views

comparing two decaying sequences

I have two sequences: $p_0, p_0 \rho_1, p_0 \rho_2, ...$ and $q_0, q_0 \gamma_1, q_0 \gamma_2, ...$. Both sequences have infinite number of terms, and both sequences sum to 1 each. All terms ...
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0answers
34 views

Limit Comparison Test: Is there any intuition behind it?

The Limit Comparison Test: Let $ a_n > 0 $ and $ b_n > 0 $ for all $ n \geq N $, $ N $ an integer. a) If $ \lim_{n \to \infty} a_n/b_n = c > 0 $, then $ \sum_{n = 1}^{\infty} a_n $ ...
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3answers
33 views

Convert recurrent formula with polynomial term / parameter to explicit formula

So, I know how to convert to explicit formulas things like the Fibonacci sequence cause it only consists of $a_n$ like this: $$a_{n} = a_{n-1} + a_{n-2}$$ However my problem is I've encountered a ...
1
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2answers
20 views

Divergence of series by comparison test with inverse elements

The title is preliminary and should be changed if anyone has a better idea how to express this. This is the series in question: $$\sum_{n=1}^{\infty} \frac{n+4}{n^2-3n+1} := \sum a_n$$ What feels ...
2
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1answer
30 views

What does $(a_n)_n \in A^{\mathbb{N}}$ mean?

What does $(a_n)_n \in A^{\mathbb{N}}$ mean? What kind of sequence is that? How does the indexing work? What's the $A$ to natural numbers power?
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3answers
28 views

Proving convergence or divergence of a series

How do I prove convergence or divergence of this series? I can't prove that it diverges using the divergence test (sequence converges to zero). Thanks in advance!
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1answer
61 views

How to solve this recursion relation?

Suppose: $$2k(n-k)a_k=n(n-1)+(n-k)(k+1)a_{k+1}+(n-k)(k-1)a_{k-1}$$ where $k=1, 2, ..., n-1$ and $a_n=0$, how to derive $a_k$? I tried to find pattern $a_1-a_2=n/2$; $a_2-a3=n(2n-1)/6(n-2)$, it ...
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6answers
2k views

Does an increasing sequence of reals converge if the difference of consecutive terms approaches zero?

If $a_n$ is a sequence such that $$a_1 \leq a_2 \leq a_3 \leq ...$$ and has the property that $\space$$a_{n+1}-a_n \longrightarrow 0$, Then can we conclude that $a_n$ is convergent? $$\space$$ I ...
1
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1answer
40 views

(Proof verification) Showing that the two $\limsup$ definitions are equivalent

I have been trying to prove that the two definitions of $\limsup$ are equivalent. I'd appreciate a lot if someone could verify my attempt! Thanks in advance! Here are the two definitions: ...
1
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1answer
47 views

A difficult limit -> problem in getting out of the form $\infty \times 0$

Let $(x_n)$ be a real sequence which converges to $l$. Moreover we have : $\mid x_n -l\mid \leq C \mid x_{n-1}-l\mid, C \in ]0,1[$ and : $\lim_{n \to \infty} \frac{x_{n+1}-l}{x_n-l} = k \in ]0,1[$. ...
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5answers
265 views

Finding limit of a (Laurent?) series

I've been practicing series for my upcoming Calculus 1 exam, and I've stumbled upon this one: $1 + \frac{1}{1 + 2} + \frac{1}{1 + 2 + 3} + ... + \frac{1}{1 + 2 + 3 + ... + n}$ The task is to find the ...
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0answers
24 views

Discussing the convergence of a sequence $x_n$ [on hold]

If $k, x_1$ are positive, and $x_{n+1}$ = $\sqrt {k + x_n},$ discuss the convergence of the sequence $x_n$, according to whether $x_1$ is less than or greater than $\alpha$, the positive root of $x^2 =...
2
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1answer
57 views

Proving Binomial identity involving algebraic expression

How did i prove $$\frac{\binom{n}{0}}{x}-\frac{\binom{n}{1}}{x+1}+\frac{\binom{n}{2}}{x+2}-\cdots \cdots +(-1)^n\frac{\binom{n}{n}}{x+n}=\frac{n!}{x(x+1)(x+2)\cdots (x+n)}$$ what i try $$\sum^{n}_{...
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1answer
77 views

Evaluate this question based on series and limits.

For $a \in \mathbb R,a≠-1$ $$\lim_{n\to\infty}\frac{1^a+2^a+\cdots +n^a}{(n+1)^{a-1}[(na+1)+(na+2)+(na+3)+\cdots+(na+n)]}=\frac{1}{60}$$ Then find the values of $a $. I tried to solve this ...
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1answer
33 views

Calculate infinite series with natural logarithm [on hold]

Please calculate next expression. $$\sum_{k=0}^{\infty}\frac{(-1)^k(ln(4))^k}{k!}$$
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0answers
32 views

Find the sum of this sequence [on hold]

Is it possible to find the sum of the sequence below? Explain why or why not. $$ \begin{cases} a_1=10&\\ a_n=3+a_{n-1}&\text{if $n\geq 2$} \end{cases}. $$
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0answers
44 views

Solution verification: Writing a sum in the Cantor normal form

Can someone please help with the following problem. I need to write the following sum in Cantor normal form: $$\sum_{i ∈ ω\cdot2} \sum_{j ∈ i} (i+j)$$ The result I'm getting is $$ω^2 + w,$$ so I ...
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2answers
66 views

Do these polynomials with harmonic number-related coefficients lie in some particular known class?

I've generated a set of univariate polynomials ($b=1,2,\ldots$) in $v$ of degree $b-1$. The constant term and the coefficient of $v^{b-1}$ is simply $H_b$, the $b$-th harmonic number. The ...
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3answers
68 views

Convergence of the series for $a \in \mathbb R$ $\sum_{n=1}^\infty\sin\left(\pi \sqrt{n^2+a^2} \right)$

Convergence of the series for $a \in \mathbb R$ $$\sum_{n=1}^\infty\sin\left(\pi \sqrt{n^2+a^2} \right)$$ I saw this problem in a calculus book and it gave a hint that says HINT First show that $$\...
2
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1answer
63 views

I didn't understand this recurrence relation solution.

Recently, i was trying to solve this recurrence relation $$ a_{n+4} = \frac{-\alpha(x)}{(n+4)} \cdot a_{n+3} +\frac{-\beta(x)}{(n+3)\cdot (n+4)} \cdot a_{n+2} $$ But i can't solve for $a_n$ I've ...
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0answers
35 views

Discuss the convergence of a sequence [duplicate]

If $k, x_1$ are positive, and $x_{n+1}$ = $\sqrt {k + x_n},$ discuss the convergence of the sequence $x_n$, according to whether $x_1$ is less than or greater than $\alpha$, the positive root of $x^2 =...
1
vote
2answers
77 views

Find $\lim_{n\rightarrow\infty}\sqrt[n]{5+n^2}$ using $\varepsilon-N$ language

Use an $\varepsilon-N$ argument to find and prove $\lim_{n\rightarrow\infty}\sqrt[n]{5+n^2}$. Try some variations of your own. I think that the limit is $1$, since the limit as $n$ tends to infinity ...
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0answers
19 views

Find extrema and roots of a function from its Fourier series expansion

That might be a dumb question, but here it comes. Say we have a function defined by its fourier series expansion as follows : $$\forall x\in\mathbb{R}, f(x)=\sum\limits_{n=1}^{\infty}A_n\cos\left(2\...
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2answers
39 views

A sequence with infinite number of limit points.

In my real analysis class, we have to determine whether or not the following is true. "There exists a sequence of real numbers that has infinite number of limit points." It then seemed to be true and ...
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0answers
26 views

Upper and lower bounds for a finite sum [duplicate]

Find upper and lower bound for the following finite sum: $\frac{1}{1} + \frac{1}{2^3} + \frac{1}{3^3} + ··· + \frac{1}{n^3}$ My attempt: Using the integral test: we know that $\frac{1}{1} + \frac{...
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0answers
22 views

Linear convergence of sequence

I have the following exercise but it seems to me that this is false : Let $(x_n)_{n \in \mathbb{N}}$ by a sequence of real numbers and $x^* \in \mathbb{R}$. We say that the sequence $(x_n)$ ...
0
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1answer
36 views

convergence of an infinite product $\prod_{j=1}^{n}{(0.5+ \frac{1}{\pi}\arctan(jx))}$

Does the following infinite product converge as $n \rightarrow \infty$ : $$ \prod_{j=1}^{n}{(0.5+ \frac{1}{\pi}{\arctan(jx)})} $$ If yes, then what is the limiting value? I encountered the above ...
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0answers
32 views

Upper and Lower bound of a finite sum

Find upper and lower bound for the following finite sum: $$\frac{1}{1} + \frac{1}{2^3} + \frac{1}{3^3} + ··· + \frac{1}{n^3} $$ My attempt is: Using the integral test: we know that $\frac{1}{1} + ...
1
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1answer
26 views

Convergence of the sum of iid scaled by $n^\alpha$

I am interested in the convergence of the sequence $\mathbb{P}(|X_1+...+X_n|/n^\alpha<z)$ where $z>0$, $\{X_n\}_n$ is an i.i.d. sequence with mean zero and finite variance. I can easily prove ...
0
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1answer
57 views

find the upper and lower bound for a finite sum

Find upper and lower bound for the following finite sum $1/(1 + 1^3)+1/(1 +2^3)+1/(1 + 3^3) + ··· + 1/(1 + n^3)$ My attempt: $1/(1 + 1^3)+1/(1 +2^3)+1/(1 + 3^3) + ··· + 1/(1 + n^3)$ = $\sum_{i=1}^n ...
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0answers
31 views

Another limit hinting at Riemann sums (but this is a misleading hint)

$\lim\limits_{n\to\infty}{\frac{1}{n} {\sum\limits_{k=1}^{n}{\frac{k}{k^2+1}}}}$ Again, this is not a Riemann sum and I have no idea how to find the sum explicitly. The only thing that comes to mind ...
1
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1answer
34 views

Help with convergence tests for series

I have a few questions to ask about series and convergence tests. I have been struggling to study everything fully and if someone can give me advices I will be really thankful.This is what I know so ...
0
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1answer
38 views

Evaluating a sum (alternating binomial series with odd denominators) [duplicate]

How do I evaluate the following sum (for some positive integer $m$)?: $$S=\sum_{k=0}^{m}{{m \choose k}\frac{(-1)^k}{2k+1}}$$ After expanding it looks like: $$S={m \choose 0}-\frac{1}{3}{m \choose 1} + ...
1
vote
3answers
64 views

Proving that $\lim_{v\rightarrow \infty}\left[\frac{v^2}{v^3 + 1} + \frac{v^2}{v^3 + 2} +\cdots + \frac{v^2}{v^3+v}\right]= 1 $

I wonder if my solution that $\lim_{v\to\infty}\left[\frac{v^2}{v^3 + 1} + \frac{v^2}{v^3 + 2} + \cdots + \frac{v^2}{v^3+v}\right]= 1 $ is correct. $$\frac{v^2}{v^3 + 1} + \frac{v^2}{v^3 + 2} + ... + ...
0
votes
0answers
15 views

Convergence of $u_{n+1} = (\frac{n}{n+2} - \frac{l(l+1)}{(n+1)(n+2)}) x^2 u_{n}$ at $x = 1$.

This infinite series converges for $x \in (-1,1)$ $$u_{n+1} = (\frac{n}{n+2} - \frac{l(l+1)}{(n+1)(n+2)}) x^2 u_{n}$$ Which can be easily determined using ratio test. But how can we test for ...
0
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2answers
39 views

Existence of a Sequence of strictly positive powers of (1/2) whose sum equals 3/8.

I am writing a proof for my measure theory course that seems to boil down to the existence of a sequence of the form $$ \left\{\frac{1}{2^{n_1}}, \frac{1}{2^{n_2}}, \ldots \right\} $$ with $n_k \in \...
3
votes
5answers
49 views

Find if terms are terms of the same arithmetic progression

Is it possible that numbers $\frac{1}{2}, \frac{1}{3}, \frac{1}{5}$ are (not necessarily adjacent) terms of the same arithmetic progression? Hint: Yes. Try $\frac{1}{30}$ as a difference. I was ...