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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
16 views

I was told that I couldn't “pull the limit in”. Tell me exactly how I'm messing up, please!

So, the problem that we were solving was The limit as n approaches infinity of (n/(n+1))^n (picture for clarity) To figure out whether the series converged or diverged, after simplification, I asked ...
3
votes
4answers
73 views

Does the series $\sum_{n=1}^\infty\frac{(2n)!}{2^{2n}(n!)^2}$ converge or diverge.

$$\sum_{n=1}^\infty\frac{(2n)!}{2^{2n}(n!)^2}$$ Can I have a hint for whether this series converges or diverges using the comparison tests (direct and limit) or the integral test or the ratio test? ...
0
votes
0answers
35 views

Finite sum of numbers

I was wondering if there is some chance to find a sequence of numbers $b_i,\,i=1,\ldots,n,$ such that the following conditions hold simultaneously: $\sum_{i=1}^n b_i=O(n)$ and $\sum_{i=1}^n b^2_i=O(1)$...
2
votes
0answers
25 views

Periodicity criteria for $f'(x)=P^{m}(f(x))$

$$f'(x)=P^{m}(f(x))$$ ,where $P(x)$ is a polynomial, m is a real number. $x\in\mathbb{C}$ and $f:\mathbb{C}\to\mathbb{C}$ I would like to find a criteria to define if $f(x)$ is a periodic function ...
-2
votes
1answer
25 views

Convergent Sequences question [on hold]

A sequence is bounded and non monotonic, can it be convergent? Why or why not?
1
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1answer
21 views

complex sum of reciprocal decimal expansions terminating in odd digits

Let bn be the sequence of all positive integers such that the decimal expression for 1/bn terminates in an odd digit: 1, 2, 4, 8, 10, . . . (For instance, 3 is not included because 1 3 = 0.33 . . . ...
0
votes
1answer
37 views

Is the series $\sum_{n=1}^{\infty} \frac{\sin nx}{n}$ is uniformly convergent on $[0,1]$? [duplicate]

Is the series $\sum_{n=1}^{\infty} \frac{\sin nx}{n}$ uniformly convergent on $[0,1]$? My work : since $|\sin nx|\le 1$ so $\sum_{n=1}^{\infty} \frac{\sin nx}{n} \le \sum_{n=1}^{\infty} \frac{1}{n}...
0
votes
1answer
35 views

Show that $\sum_{n=1}^{2^k-1} 1/n^p < \sum_{n=0}^{k-1} (1/2^ { p-1})^n)$

I was asked to show that the p-series $\sum_{n=1}^{\infty} 1/n^p$ is convergent without using the integral and limit test. Also, I need to show that $$\sum_{n=1}^{2^k-1} \frac{1}{n^p} < \sum_{n=0}^{...
0
votes
2answers
54 views

$\sum\limits_{n=1}^{\infty} \frac{1}{n\sqrt[n]{n}}$ converge? [on hold]

Does the series $\sum\limits_{n=1}^{\infty}\frac{1}{n\sqrt[n]{n}}$ converge?
1
vote
1answer
54 views

Proving that $\sum\limits_{n=1}^∞\frac{a_n}{s_n^2}$ converges

Let $a_n>0$ and given that $\sum\limits_{n=1}^\infty a_n$ divergent and $s_n =\sum\limits_{k=1}^na_k$. For all $n \ge 2$, prove that $\sum\limits_{n=1}^∞\dfrac{a_n}{s_n^2}$ converges. Proof: ...
0
votes
1answer
38 views

What is this series of numbers called?

Let's say $N$ is the initial number. $x_0 = \frac{N}{4}$ $x_1 = \frac{N - x_0}{4}$ $x_2 = \frac{x_0 - x_1}{4}$ $x_3 = \frac{x_1 - x_2}{4}$ and so on... What is this series called? Is there a way to ...
1
vote
2answers
24 views

What is the explicit formula for the general term of the sequence?

A sequence $\{a_n\}_{n \ge 1}$ is defined recursively by $$a_0 = 1, a_1 = 1$$ $$a_n = 5a_{n-1}-6a_{n-2}, \text{ for } n \ge 2 $$ Find an explicit formula for the general term $A_n$. So, I want to ...
2
votes
5answers
60 views

Does $\sum_{n=1}^{\infty}\frac{1}{\sqrt{1}+\sqrt{2}+\cdots+\sqrt{n}}$ converge?

Looking for exercises for my students, I stumbled upon this series $$\sum_{n=1}^{\infty}(-1)^n\frac{1}{\sqrt{1}+\sqrt{2}+\cdots+\sqrt{n}}$$ which can be easily seen to be convergent thanks to the ...
0
votes
1answer
19 views

Two sums, the relation between them

What is the relation between sum? Is something interesting? Is it possible taking square or solving set of equation calculate "non squared" sum? Thank you for your explanation. $$\sum\limits_{n=2}^{\...
1
vote
1answer
23 views

Show $(a_n)_{n=1}^\infty$ converges to $L \in \mathbb{R}$ iff the limit superior and limit inferior converge to $L$

Show $(a_n)_{n=1}^\infty$ converges to $L \in \mathbb{R} \iff lim\ sup\ a_n = lim\ inf\ a_n = L$. Direction: $\implies$ Suppose $(a_n)$ converges to $L \in \mathbb{R}$, show $lim\ sup\ a_n = lim\ inf\...
1
vote
1answer
18 views

Prove the limit superior of a bounded sequence converges

Let $(a_n)_{n=1}^\infty$ be a bounded sequence and $b_n = \sup\{a_k\ |\ k \geq n\}$. Prove $b_n$ converges. This is the limit superior of $(a_n) := \limsup\ a_n$. Wanted to see if my proof made sense....
4
votes
2answers
43 views

Evaluation of an infinite series

Consider the series $$\sum_{n=2}^{\infty} \frac{(-1)^{n} \, (n^2 - n +1)^2}{(n-2)! + (n+2)!}.$$ The series converges to some value near $0.12122103\cdots$. The series may also be seen in the form $$\...
2
votes
1answer
46 views

Verify proof that $x_n = \sum_{k=1}^n {k \over 2^k}$ is bounded and find its supremum and infinum

The problem I'm solving states: Let $n\in \mathbb N$ and $x_n$ be a sequence: $$ x_n = \sum_{k=1}^n {k \over 2^k} $$ Prove $x_n$ is bounded and find $\sup\{x_n\}$ and $\inf\{x_n\}$ Let $S_n$ ...
6
votes
1answer
53 views

Series of Nested Radicals

I can't seem to find a way, squaring the expression would make more terms and would make it harder, I guess there must be something to do with the first and last terms as they sum to 100? or maybe ...
0
votes
2answers
41 views

If {$|a_n|$} is divergent, can {$a_n$} be convergent? If {$a_n^2$} is convergent, can {$a_n$} be divergent?

Can anyone help me prove these problems? If {$|a_n|$} is a divergent sequence, can {$a_n$} be a convergent sequence? If {$a_n^2$} is a convergent sequence, can {$a_n$} be a divergent sequence?...
2
votes
2answers
37 views

The series $\sum_n^\infty a_n^p$ where $\{a_n\}_{n=1}^\infty$ is a convergent, strictly positive sequence

Suppose that $\{a_n\}_{n=1}^\infty$ is a sequence of strictly positive numbers and that $\sum_n^\infty a_n=A$ is a convergent series. Suppose that $p >1$.What can you say about the series $\sum_n^\...
-1
votes
1answer
33 views

Archer duel problem

Two archers take alternating turns trying to hit the opposing archer. As soon as the shooting archer hits the other, the shooting archer wins. Both have a specific probability p to hit the other. p1 ...
2
votes
1answer
31 views

If $\lim_{n \rightarrow \infty }s_n = s$, is it true that $\lim_{n \rightarrow \infty} \frac{1}{n}\sum_{k=1}^n s_k =s $ as well? [duplicate]

If $\{s_n\}$ is a sequence of positive real numbers such that $\lim_{n \rightarrow \infty }s_n = s$, is it true that $\lim_{n \rightarrow \infty} \frac{1}{n}\sum_{k=1}^n s_k =s $ as well?
0
votes
1answer
19 views
1
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2answers
29 views

Showing a series converges absolutely almost everywhere

Let $f:\mathbb{R}^m\rightarrow\bar{\mathbb{R}}$ be a Lebesgue integrable function with $\int |f|>0$. Show that the infinite series $\sum_n\frac{f(n\vec{x})}{n^p}$ converges absolutely almost ...
0
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0answers
16 views

Distance between any arbitrary sequence and an exponential sequence

Let $b \in (1,c)$ for some $c$. Now, we ask someone to choose a sequence $\{a_t: t \geq 0 \}$ with knowing what $c$ is, but without knowing what $b$ is. Then, we measure $$f_T = \sum_{t=0}^T (a_t - ...
1
vote
1answer
33 views

Determine whether $\sum_{k=1}^{\infty}\frac{k+2}{\sqrt{k^5+4}}$ converges or diverges

Question Determine if $$\sum_{k=1}^{\infty} \frac{k+2}{\sqrt{k^5+4}}$$ converges or diverges. I'm looking for a proof that this sum converges in a simpler way than I've shown. My (ugly)...
0
votes
1answer
29 views

If part of an infinite series diverges, does the whole series diverge?

If you can write some infinite series $S_1$ as the sum $S_2+S_3$, is it always the case if atleast one of $S_2, S_3$ diverges that $S_1$ diverges?
0
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1answer
32 views

Why ${a_n^{1/n}}$ converge to 1, if ${a_n}$ converge to a a positive real.

How you prove that. Intuitive is obvious since if $a$ is bigger than 1, then you use the classical proof and if not you can bounded the number with two fractions of the form $1/n$, and the divisor ...
0
votes
0answers
21 views

Suppose that $lim_{n\rightarrow\infty}\ a_n = L$. Show that $lim_{n\rightarrow\infty} \frac{a_1 + a_2 + \dots + a_n}{n} = L$ [duplicate]

Suppose that $lim_{n\rightarrow\infty}\ a_n = L$. Show that $lim_{n\rightarrow\infty} \frac{a_1 + a_2 + \dots + a_n}{n} = L$ Would appreciate a hint here. Not quite seeing the solution (Tried ...
1
vote
1answer
39 views

The convergence of $ \sum_{i=1}^n \frac{1}{p(n)}$ when p(n) is a polynomial of degree bigger than 1

How can I prove the convergence of $\sum_{i=1}^n \frac{1}{p(n)}$, I know the bigger term is going to dominate and since $\sum \frac{1}{n^2}$ converge then also the first sum however I dont know how ...
0
votes
2answers
15 views

Help proving inequality by induction with recurrent sequence?

Problem For a sequence, $u_n$ , $u_1=u_2=1$ and $u_{n+2}=u_{n+1}+u_n$ Using induction, prove $u_n<2^n$ So, I'm having trouble working through this. I've tried coming up with a conjecture for $...
1
vote
2answers
45 views

Radius of convergence of $\sum \frac{z^n}{n \log n \log\log n}$

I'm trying to find the radius of convergence (along with consideration of boundary behavior) of $$\sum S_n=\sum_n^\infty \frac{z^n}{n \log n \log\log n}$$ I know that series of the form $\sum \frac{1}...
1
vote
0answers
25 views

How to generate Fourier invariant closest norm sequence?

I am not sure if it is a useful question but would like to know if possible. Consider I have a random numbers $A=a_1, a_2,,a_n$. Now suppose, I do Fourier Transform of A (for example use fft command ...
0
votes
0answers
11 views

Proving that the sum of the difference of square roots of partial sums diverges

Let $(a_n)_{n∈N}$ be a sequence of positive numbers which tends to zero but such that $\sum_{n=1}^\infty a_n$ diverges. Let $(A_n)_{n∈N}$ be the sequence of partial sums $A_n=\sum_{k=1}^n a_k$,and let ...
0
votes
1answer
61 views

Is this sum constant for n?

Hi I can prove that this sum is constant in $n\in \mathbb{N}$. However my proof is very long (a few pages with probability involved). Does anyone see a simple proof. The sum in question is (a q-series ...
0
votes
3answers
49 views

Showing that $\lim_{n\to\infty}\sum_{k=n}^{2n-1}\frac{1}{k} = \ln 2$ [on hold]

Show that $$\lim_{n\to\infty}\sum_{k=n}^{2n-1}\frac{1}{k} = \ln 2.$$ I have no clue, any help would be appreciated!
3
votes
1answer
59 views

What does $\lim \inf _ { n \rightarrow \infty }$ mean?

I'm new into Mathematical Analysis, and my textbook says: Consider the series: $\frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 2 ^ { 2 } } + \frac { 1 } { 3 ^ { 2 } } + \frac { 1 } { 2 ^ { ...
1
vote
4answers
38 views

Limit of sequence $x_n=\frac{1^p+2^p+…+n^p}{n^{p+1}}$

The limit the following sequence is $\frac{1}{p+1}$. I have tried to prove it, but I don't success. Could you help me? $$x_n=\frac{1^p+2^p+...+n^p}{n^{p+1}}$$ $$p\in\mathbb{N}$$ Thanks in advance!
1
vote
1answer
20 views

Is $l_p$ a closed subspace of $c_0$?

I would think it is, but I am not sure. In an attempt trying to prove it, I take a series $(a_n)_m:\mathbb{N}\rightarrow l_p$, which is convergent in $c_0$. My goal is to show that $\lim_{m\...
1
vote
4answers
36 views

Identifying whether the sequence $2n+(-1)^n$ is monotonically increasing/decreasing and if it is limited.

(First of all, sry for some spelling and similar mistakes as English is not my native language.) I tried to prove, that the sequence $2n+(-1)^n$ is monotonically increasing, that's what I assume from ...
-1
votes
0answers
37 views

Question about limits?

For each $n \in \mathbb{N}$, let $A_n$ be a finite set of numbers. Assume that for each $n$ and $m$, such that $n \ne m$, the sets $A_n$ and $A_m$ are disjoint. Define $$ f(x) = \begin{cases} 1/n, &...
0
votes
1answer
33 views

How to check the convergence of $\sum_{n=1}^{\infty} \frac{n^{n/2}}{n!}$ [on hold]

How do I know the nature of the $$\begin{align} \sum_{n=1}^{\infty} \frac{n^{n/2}}{n!} \end{align}$$ I have tried the ratio test and got nowhere.
0
votes
2answers
47 views

How to compute the value of an infinite series

How do I compute this series$$\sum_{n=3}^{\infty}\frac{4}{n^2 - 4}$$ I know I should consider the k-th partial sum, which is $$\sum_{n=3}^{k}\frac{1}{2n - 8}-\frac{1}{2n+8}$$ By observing which terms ...
2
votes
3answers
47 views

Check proof that $\prod_{k=1}^n(1+{1\over a_k})$ is bounded if $a_{n+1} = (n+1)(a_n + 1)$ and $a_1 = 1$, $n\in \mathbb N$

Let $n \in \mathbb N$ and: $$ \begin{cases} a_1 = 1 \\ a_{n+1} = (n+1)(a_n + 1) \end{cases} $$ Prove that $$ x_n = \prod_{k=1}^n\left(1+{1\over a_k}\right) $$ is a bounded sequence. Obviously ...
0
votes
1answer
41 views

Prove that $\sum_{n=1}^{\infty}\frac{x^2}{1+n^3x^4}$ converges uniformly for $x\in \mathbb{R}$.

Prove that $$\sum_{n=1}^{\infty}\frac{x^2}{1+n^3x^4}$$ converges uniformly for $x\in \mathbb{R}$. My try: for $x=0$, of course it converges. for $x\neq 0$ , then $\frac{x^2}{1+n^3x^4}\le \frac{1}{n^...
0
votes
0answers
13 views

How to rewrite $\sum_{k\in \mathbb Z} e^{-2\pi i sk } g\left(\frac{t-k}{2}\right)$?

Assume that $g\in \mathcal{S}(\mathbb R)$ (Schwartz class) function so that everything below make sense. Can we expect following $$\sum_{k\in \mathbb Z} e^{-2\pi i sk } g\left(\frac{t-k}{2}\...
0
votes
0answers
49 views

prime revealing function

I would like to ask how can one prove that if $$f(n,x) = \prod_{i=1}^{n}sin(\frac{1}{p_i}x\pi) $$ where $p_i$ is the i-th prime number then foreach $$n,x \in N; x>p_n;x<(p_n+2)^2;f(n,x)\neq0 ...
4
votes
0answers
41 views

Assume $x_n>0$,$x_n+\dfrac{4}{x_{n+1}^2}<3.$ Prove $\lim\limits_{n \to \infty}x_n$ exists and evaluate it.

My Solution Notice that $$x_n+\dfrac{4}{x_{n+1}^2}<3=3\sqrt[3]{\frac{x_n}{2}\cdot\frac{x_n}{2}\cdot\frac{4}{x_n^2}}\leq \frac{x_n}{2}+\frac{x_n}{2}+\frac{4}{x_n^2}=x_n+\frac{4}{x_n^2}.$$ This ...
0
votes
0answers
21 views

show that $(x_n)_{n\geq0}$ and $(y_n)_{n\geq0}$ are l.i. in $Fib$ => $(x_0,x_1);(y_0,y_1)$ are l.i. in $\mathbb{R}^2$

Let $Fib=\{(x_n)_{n\geq0}|x_{n+2}=x_{n+1}+x_n, x_n\in\mathbb{R}\}$. Determine a base in Fib. My attempt: It is easily to be seen that $Fib$ is a vectorial space over $(\mathbb{R}, \oplus,\odot)$. ...