For questions about recurrence relations, convergence tests, and identifying sequences

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1answer
16 views

Alternative representation of time series

In a paper I am reading, it refers to the following time series model: $$ Y_t=\rho Y_{t-1}+e_t $$ Where $ \lvert\rho\rvert < 1$ It goes on to say that this process can be represented in the ...
-1
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1answer
50 views

Show that if $\lim_{n\rightarrow\infty}z_n=0$ then $\lim_{n\rightarrow\infty}(1+\frac{z_n}{n})^n$ [on hold]

Can you help me with the following problem. I dont have any idea how to start. Prove that if $\lim_{n\rightarrow\infty}z_n=0$ then $\lim_{n\rightarrow\infty}(1+\frac{z_n}{n})^n=1$.
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3answers
47 views

how to find sum of the given series

How to find sum of the following series: $1+\dfrac{1}{3}\dfrac{1}{4}+\dfrac{1}{5}\dfrac{1}{4^2}+...$ The general term is $u_n=\frac{1}{2n+1}\frac{1}{4^n};n\geq 1$ Any help
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4answers
50 views

How to check the convergence of $\sum_{n=1}^\infty\frac{\ln(n^2+n+1)}{n^{3/2}}$

There is an example of the Limit Comparison test on my textbook, and it finds the convergence of this series: $$\sum_{n=1}^\infty\frac{\ln(n^2+n+1)}{n^{3/2}}$$ It starts off with the limit ...
-1
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1answer
39 views

Proving a sequence is Null, Help!

I have this question: Use the definition of a null sequence to prove that the sequence $\{a_n\}$ given by $a_n = \dfrac{2}{2n^2 -3}, n = 1, 2, \dots ,$ is null. So I know that we want to show for ...
1
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0answers
35 views

Test $\sum_{n=1}^{\infty}\Big(1-n \sin(\frac{1}{n})\Big)$ for convergence with asymptotic comparison

Suppose we want to test the following series for convergence $$\sum_{n=1}^{\infty}\Big(1-n \sin(\frac{1}{n})\Big)$$ I have found a solution that uses the asymptotic comparison test and uses the ...
5
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0answers
208 views

Successive ratios of a sequence, is this limit true?

The natural numbers $1,2,3,4,5....$ can be calculated as the row sums of the triangle $T(n,k)$ equal to $1$ if $n \geq k$ and $0$ otherwise: $$\displaystyle T = \left(\begin{matrix} ...
2
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2answers
17 views

Finding the nth term in a recursive coupled equation.

I'm probably missing something simple, but if I have the recursive sequence: $$ a_{i+1} = \delta a_i+\lambda_1 b_i $$ $$ b_{i+1} = \lambda_2 a_i + \delta b_i $$ how would I find a formula for $a_n$, ...
2
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1answer
75 views

Is this series: $\sum_{n=1}^{\infty}{{1 \over n} \cos{(n)} \sin{(nx)}}$ convergent?

How can I show that the following series is convergent or divergent ? $$\sum_{n=1}^{\infty}{{1 \over n} \cos{(n)} \sin{(nx)}},x\in \mathbb{R}$$ I want to use Abel-Dirichlet criteria. I've ...
1
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1answer
34 views

How to find $ \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)^2}$?

Let $f$ be a $2\pi$-periodic function whose restriction on $[-\pi, \pi]$ is $f(x)_{[-\pi, \pi]} = |x|$ It is easy to see that its fourier series converges uniformily to $f$ and is $$f(x) = \frac \pi2 ...
0
votes
1answer
31 views

Regarding sup and inf of a continuous function

Suppose $f:\mathbb R\to \mathbb R$ be a continuous function such that $\lim\limits_{x\to \infty}f(x)=0=\lim\limits_{x\to -\infty} f(x)$. Then I want to show that $f$ is bounded and attains at least ...
0
votes
2answers
45 views

Is $\sin\left[{\pi \cdot \frac {1}{\sqrt{n^2+1}+n}}\right]$ decreasing?

How can I show that $$\sin\left({\pi \cdot \frac {1}{\sqrt{n^2+1}+n}}\right)$$ is decreasing for $n>1$? I think I have to show that the expression from inside the $\sin$ expression is between $\pi ...
2
votes
3answers
53 views

Prove that {$r_n$} satisfies $r_n = 7r_{n-1} - 10r_{n-2}, n \geq 2$

Problem: For the sequence $r$ defined by $$r_n = 3 \cdot 2^n - 4 \cdot 5^n, \ \ \ n \geq 0$$ Prove that {$r_n$} satisfies $$r_n = 7r_{n-1} - 10r_{n-2}, \ \ \ n \geq 2$$ Can this problem be ...
0
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1answer
47 views
2
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4answers
117 views

The convergence of $\sum_{n=2}^{\infty} {1\over \ln(n!)}$

Check the convergence of $\sum_{n=2}^{\infty} {1\over \ln(n!)}$. I tried D'Alembert's test... Cauchy's test seems too intricate... I can't seem to understand what I should do here...
13
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1answer
351 views

Another Ramanujan's formula dealing with $\coth^{2}(5\pi)$

Ramanujan mentions in one of his letters to Hardy that $$\frac{1^{5}}{e^{2\pi} - 1}\cdot\frac{1}{2500 + 1^{4}} + \frac{2^{5}}{e^{4\pi} - 1}\cdot\frac{1}{2500 + 2^{4}} + \cdots = ...
1
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1answer
31 views

Differentiating under the summation

I saw on the Wikipedia page for differentiation under the integral that it could also be applied to summations. Here is the link: ...
1
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0answers
58 views

What is an elegant way to express $(-1)^k$

In computation of series, a lot of times you will find a term $(-1)^k$ jutting out in an otherwise easy to remember expression. Is there some interesting way to write $(-1)^k$ that may help in ...
9
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3answers
294 views

Finding 1000th 5-smooth number

Smooth numbers are natural numbers that are products of only small prime numbers. They have some applications in cryptography. A number is 5-smooth if its only prime factors are $2,3$ or $5$. ...
13
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2answers
2k views

Do these series converge to the Mangoldt function?

Jeffrey Shallit formulated this recurrence for me: $\displaystyle T(n,1)=1, k>1: T(n,k) = \sum\limits_{i=1}^{k-1} T(n-i,k-1)-\sum\limits_{i=1}^{k-1} T(n-i,k)$ which is the lower triangular array ...
4
votes
1answer
54 views

Finding the limit of a sequence of sequences

Take any $\bar{r} \in \mathbb{R}$ with $\bar{r}>0$. Assume that $f : \mathbb{R} \rightarrow \mathbb{R} $ is continuous. Assume that for all $r\in [0,\bar{r})$, there exists a strictly decreasing ...
0
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0answers
32 views

Solving a recurrence relation using the substitution method

Consider the recursive function $f(n)=3f(n/4)+2n $, $f(16)=32.$ Where n is always a power of 4 greater than 16. We must find a closed form utilizing substitutions. So, after one substitution, f(n) ...
1
vote
2answers
75 views

Determine if the following series is convergent or divergent $\sum_{n=1}^{\infty}\frac{ \sin{nx}}{n}$

I've been doing exercises with series for some time and now I've got an exercise that I'm supposed to solve using Abel or Dirichlet criteria. The problem is : Determine if the following series ...
1
vote
3answers
71 views

calculate radius of convergence

Let $\{a_n\}_{n=0}^{\infty}$ be sequence such that $$a_1 = a_0 = 1$$ $$a_{n+1}=a_n+ a_{n-1}$$ show that the radius of convergence of $\sum\limits_{n=0}^{\infty \:}a_nx^n$ is ...
1
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2answers
27 views

Converges or diverges: $\sum_{n=1}^{\infty}\left [\arctan{\frac{(-1)^n}{n}}+{(-1)^n\over n^2}\right]$

How can I show that the following series converges or diverges ? $$\sum_{n=1}^{\infty}\left [\arctan{\frac{(-1)^n}{n}}+{(-1)^n\over n^2}\right]$$ $\sum_{n=1}^{\infty}\left ...
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votes
1answer
56 views

Maximising $\log_{2015} ( a_{2015} ) - \log_{2014} ( a_{2014} )$ given the properties of the sequence [closed]

Let $a_1, a_2, a_3 , \ldots$ be a sequence of positive real numbers such that For all positive integers $m$ and $n$ we have $a_{mn} = a_ma_n$, and There exists a positive real number ...
0
votes
2answers
42 views

Generalized geometric series value

Why the value of the following summation: $$1 + \sum_{k=1}^{n}\bigg(1- \frac{76}{i}\bigg)^k= \frac{i}{76}$$ is $\frac{i}{76}$? $\quad i$ is a positive constant.
9
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4answers
203 views

Ways to prove $ \int_0^1 \frac{\ln^2(1+x)}{x}dx = \frac{\zeta(3)}{4}$?

I am wondering if we can show in a simple way that $$ I=\int_0^1 \frac{\ln^2(1+x)}{x}dx = \int_1^2 \frac{\ln^2(t)}{t-1}dt = \frac{\zeta(3)}{4}. $$ Because the end result is very simple, I suspect ...
5
votes
2answers
54 views

Prove a convergent sequence has either a minimum, a maximum or both.

Let $a_n$ be a convergent sequence. Prove $a_n$ has a minimum, a maximum or both. I am being prepared for a final exam, which is why it is important to me to know that $I$ am correct in $my$ ...
3
votes
1answer
25 views

$l_2$ sequence, series with square root

I'm trying to prove that the following functional is continuous: $$\phi : \mathcal{l}_2 \ni \{x_n \} \rightarrow \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}x_{3n} - \sum_{n=1}^{\infty} \frac{1}{n}x_{2n} ...
3
votes
2answers
81 views

How to show that $ \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \ln\left(\tan(x)+\tan\left(\frac{\pi}{6}\right)\right)\tan(x)\space dx=\frac{\zeta(2)}{6} $

I was trying to prove the well known result: $$ \sum_{k=1}^\infty \frac{1}{\binom{2k}kk^2}=\frac{\zeta(2)}{3} $$ and it came down to prove the following equation: $$ ...
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2answers
34 views

Determine if it converges or diverges : $\sum_{n=1}^{\infty} \frac {2^n \cdot n!}{1\cdot2\cdots (2n-1)}\cdot \frac{1}{\sqrt{2n+1}}$

Here's the series: $$\sum_{n=1}^\infty \frac {2^n \cdot n!}{1\cdot2\cdots (2n-1)}\cdot \frac{1}{\sqrt{2n+1}}$$ Does it converge or diverge ? Thanks
0
votes
1answer
51 views

Convergence of: $\sum_{n=1}^{\infty} \frac{2+(-1)^n}{2^n+(-1)^n} $

I've tried solving this exercise but got stuck on a big expression that I could not untangle. I've obtainded the following thing: $$\lim_{n \to \infty} n\frac{2 \cdot 2^n +3 \cdot 2^n\cdot ...
3
votes
2answers
61 views

Is $\lim\sup=\sup\lim$?

Assume $(a_n(x))_{n=1}^{\infty}$ is a bounded sequence in $\mathbb R$, when $x$ is $\in\mathbb R$ and is relevant to the sequence in some way that doesn't really interest us in my question. Assume ...
1
vote
3answers
191 views

Convergence of the series $\sum_{i=1}^\infty \sqrt{2n+1}/n^2$

How does the series $\sum_{i=1}^\infty \sqrt{2n+1}/n^2$ converge? I have yet to receive a result that is not inconclusive. If you could tell me what test you used to confirm its convergence that ...
1
vote
2answers
32 views

Indeterminate form as a series

We know that $0 \times \infty$ is an indeterminate form. However, is it equivalent to $0 + 0 + 0 + \cdots$? If yes, why we do not consider $\displaystyle \sum_{n = 0}^\infty 0$ an indeterminate form? ...
0
votes
1answer
58 views

Hyperreal probability density?

I'm fairly new to the subject of hyperreal numbers and I'm wondering if there exists an infinitesimal number $a$ such that (in some reasonable sense) $$\sum_{n=1}^\infty a=1$$ ? In other words: Is ...
1
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1answer
15 views

Alternating series, even terms, factorial, boundedness

I need to determine whether there exists $M>0$ such that $$| \sum_{n=0}^{\infty} \frac{(-1)^n x_{2n}}{\sqrt{n!}}| \le M \sqrt{\sum_{n=0}^{\infty}|x_n|^2}$$ $\{x_n\} \in \mathcal{l}^2, \ \ x_n \in ...
0
votes
1answer
27 views

Lagrange inversion theorem application

Can someone give me an example of where the Lagrange inversion theorem is applied in such a way it inverts a formal series? For example, say I have $$\sum_{i>-1} a_it^i = u.$$ Can someone show ...
0
votes
4answers
37 views

Question of monotone convergent sequence

My Question is now " Prove that $(2^n+3^n)^{1/n}$ is convergent. So I proved for two method by using monotone convergence theorem and squeeze theorem such as monotone 1) bounded below since 3 is ...
3
votes
0answers
131 views

What is a better comparision series for $\sum \frac {1}{F_n}$ than $\sum_0^\infty2^{-k}$?

What is a better comparision series for $\sum \frac {1}{F_n}$ than $\sum_0^\infty2^{-k}$? By better comparison series than $\sum_0^\infty2^{-k}$ we mean a series $\sum c_k$ s.t. ...
0
votes
1answer
21 views

Construct a non-constant sequence

Let $$S^{n}_{r}=\bigl\{{\overline{x}\in\mathbb{R}^{n+1}\;:\; \|{\overline{x}}\|=r}\bigr\}$$ thn $n-$ sphere with radius $r$ where $\|{\cdot}\|$ is the usual norm in $\mathbb{R}^{n+1}$ and a point ...
0
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1answer
44 views

Sequence of positive reals

Suppose we had a sequence of positive reals such that $x_{ij}=x_i\cdot x_j$ $\frac{x_{i}}{x_{j}}, \space\space i<j$ is bounded above by some positive real. Find all possible sequences. I got ...
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0answers
15 views

Separated sequence and $\sup_n|\Im \delta_k|<\infty$. [closed]

The sequence $\{\delta_k\}_{k\in\mathbb Z}\in\mathbb C$ is said to be separated if for some $\epsilon>0$: $$|\delta_m-\delta_n|\geq \epsilon$$ whenever $m\neq n$. Prove that, if ...
0
votes
1answer
25 views

Are $\sup_n |\Re a_n|<a\in \mathbb R, \sup_n|\Im a_n|<+\infty$ and $|a_n|<a\ \forall n\in\mathbb N $ in contradiction?

Let us consider a sequence of complex numbers $a_n\in \mathbb C$. Prove that (or prove otherwise if it is not true) the (1)-(2) is in contradiction: $$\sup_n |\Re a_n|<a\in \mathbb R, \ \ \ \ \ \ ...
5
votes
1answer
238 views

What is the sum of Fibonacci reciprocals?

How can I calculate $\sum\limits_{n=1}^{\infty}\frac{1}{F_n}$, where $F_0=0$, $F_1=1$ and $F_n=F_{n-1}+F_{n-2}$? Empirically, the result is around $3.35988566$. Is there a "more mathematical way" to ...
2
votes
2answers
62 views

Geometric Series with coin tosses

Suppose you toss a coin and observe the sequence of H’s and T’s. Let N denote the number of tosses until you see “TH” for the first time. For example, for the sequence HTTTTHHTHT, we needed N = 6 ...
1
vote
1answer
24 views

My proof of: Given an adherent point, some sequence converges to it.

(I have two specific questions.) Is my proof correct? Are style, wording, and punctuation alright? $\textbf{Definition 1.}$ Let $X \subseteq \mathbb{R}$. Let $x \in \mathbb{R}$. The point $x$ ...
2
votes
2answers
55 views

How to prove that $\lim_{n\to\infty}\frac{x_1+2^kx_2+\dots+n^kx_n}{n^{k+1}}=\frac{x}{k+1}$

Knowing that $\lim_{n\to\infty}x_n=x$ I want to prove that $$\lim_{n\to\infty}\frac{x_1+2^kx_2+\dots+n^kx_n}{n^{k+1}}=\frac{x}{k+1}.$$ My guess is that we will use the Stolz–Cesàro theorem. So for ...
2
votes
4answers
614 views

Why is 11 times the 7th term of a fibonacci series equal to the sum of 10 terms?

Why is 11 times the 7th term of a fibonacci series equal to the sum of 10 terms? I was watching scam-school on youtube the other day and this number trick just astonished me. Can someone please ...