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

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8
votes
2answers
133 views

Can a series be empty?

It sounds contra-intuitive, since it won't exist without elements. However, I am thinking in empty sets, null sequences and empty lists, which indeed exists.
19
votes
2answers
500 views

Ramanujan style nested differential Equation

So I was exploring some math the other day... and I came across the following neat identity: Given $y$ is a function of $x$ ($y(x)$) and $$ y = 1 + \frac{\mathrm{d}}{\mathrm{d}x} \left(1 + ...
1
vote
2answers
2k views

Prove that the taylor series of cos(z) and sin(z) are holomorphic

I have a question on an old exam paper given as follows: If we define $$\cos(z) = 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - ... \frac{z^{2n}}{(2n)!} + ... = \sum_{n=0}^{\infty} ...
6
votes
3answers
196 views

Find the limit of $ x_n = \prod_{j=2}^{n} \left(1 - \frac{2}{j(j+1)}\right)^2$

I am stuck on the following problem: Let $x_n=(1-\frac{1}{3})^2(1-\frac{1}{6})^2(1-\frac{1}{10})^2 \ldots...(1-\frac{1}{n(n+1)/2})^2, \text{where} \space n \geq 2$. Then $\lim_{n \to ...
0
votes
2answers
710 views

prove - if a tends to L then Ma tends to ML

Suppose that the sequence $(a_n)_{n=1}^{\infty}$ is such that $a_n \rightarrow L$ as $n\rightarrow\infty$. Prove that for any $M>0, M \in \mathbb{R}$, we have that $Ma_n \rightarrow ML$ as $n ...
9
votes
4answers
207 views

a sequence $\{s_n\}$ with $\sum s_n$ convergent

what would be $\{s_n\ge0\}$ such that $$\sum_{n=1}^\infty s_n$$ converges but $$\lim_{n\to\infty}(n s_n) \neq 0$$
1
vote
2answers
67 views

Is $u_n\le(1-a)^n\forall n\in\mathbb{N}$?

Consider the sequence $\{u_n\}$ where $u_0=1,u_1=1-a$ for some $0< a < 1/4$, and $u_{n+2} = u_{n+1}-au_n$. Is $u_n\le(1-a)^n\forall n\in\mathbb{N}$?
1
vote
1answer
72 views

proof of local maximum

If we have a sequence of real numbers $[a_1, a_2,..., a_n]$, then this sequence has got a p-step local maximum at the $kth$ position, if max {1, k-p} $\le$ m $\le$ min {n, k+p}, $a_m \le a_k$. Now, ...
1
vote
1answer
113 views

Numerical Analysis converging sequence question

Show that the sequence $p=10^{-2n}$ converges to zero with order $2$. How many steps, $n$, will it take before this sequence is within $10^{-8}$ of zero? Construct a sequence that converges with ...
8
votes
1answer
216 views

Function mapping challange

For a given set $A=\{1, 2, 3, 4, \ldots, n\}$, find the number of non-constant mappings (associations ) $f$ from $A$ to $A$ such that $f(k) \leq f(k + 1)$ and $f(k) = f(f(k + 1))$. This is ...
0
votes
1answer
207 views

Relation between infinite series and integrals

Is there any relation between the definite integral and infinite series excluding the fact that an improper integral can be viewed as an infinite series whose terms are definite integrals? By ...
0
votes
2answers
87 views

Differentiation of a series of function

If $f(x) = \frac{x-1}{4}+\frac{(x-1)^3}{12}+\frac{(x-1)^5}{20}+\frac{(x-1)^7}{28}\ldots$ where, $0 <x<2$ then find the derivative of $f(x)$ Please guide how to proceed for this. This is an ...
7
votes
2answers
537 views

Summing series with factorials in

How do you sum this series? $$\sum _{y=1}^m \frac{y}{(m-y)!(m+y)!}$$ My attempt: $$\frac{y}{(m-y)!(m+y)!}=\frac{y}{(2m)!}{2m\choose m+y}$$ My thoughts were, sum this from zero, get a trivial ...
1
vote
1answer
47 views

On the limit of a Minkowski sum

Consider an open set $\mathcal{O} \subseteq \mathbb{R}^n$. I am wondering if the set $$ \mathcal{S} := \lim_{k \rightarrow \infty} \ \mathcal{O} + \frac{1}{k} \mathbb{B} $$ is open or closed. With ...
2
votes
1answer
75 views

Computation of standard series

I am stuck on the computation of the following sum: $\sum_{k=1}^\infty e^{-n^2}\cos(n)$. Simple tricks fail and also i have no idea how to fit it for Fourier series. Are there any other ways?
1
vote
3answers
1k views

Testing for convergence in Infinite series with factorial in numerator

I have the following infinite series that I need to test for convergence/divergence: $$\sum_{n=1}^{\infty} \frac{n!}{1 \times 3 \times 5 \times \cdots \times (2n-1)}$$ I can see that the denominator ...
3
votes
1answer
28 views

Given $\alpha>0$. Prove that eventually $e^{n^{\alpha}}>n$, for $n \in \mathbb{N}$

Given $\alpha>0$. Prove that there exists $N \in \mathbb{N}$ such that $e^{n^{\alpha}}>n$, for $n \in \mathbb{N}$, $n \geq N$.
2
votes
3answers
94 views

recursive sequence

Given $0<a<b$, $\forall n$ define $x_n$ as $x_1=a$, $x_2=b$, $x_{n+2}=\frac{x_n+x_{n+1}}{2}$. Show that $(x_n)$ converges and find the limit. In order to prove the convergence, I claim that ...
2
votes
2answers
252 views

Sine and Cosine Power Series

I have read that sine and cosine can be represented as power series. Power series, as I understand them, are infinite series that can be represented as: $\sum_{j=0}^{\infty} a_j (x-x_0)^j$ where ...
0
votes
1answer
40 views

Help with the following series

Can anyone help me to find the value of the following series? $\sum_{n=1}^{\infty} \dfrac{1}{1+x^{2n}}\quad $ and $\quad \sum_{n=1}^{\infty} \dfrac{1}{n^{3/2}} \left(\dfrac{ x}{1+x}\right)^n$ ...
2
votes
3answers
58 views

Series evaluated to $m$ terms, approximating the error

Given a series $\displaystyle\sum_{n=0}^\infty a_n$, how can we bound the error (which I shall denote with $R_n$) when we evaluate it to $m$ terms? $$\sum_{n=0}^\infty a_n \approx \sum_{n=0}^m a_n$$ ...
0
votes
1answer
98 views

Partial fraction

How to expand the following expression $\frac{1}{(x^n-1)(x-1)}$ in partial fraction, I think it will be rewritten in terms of geometric series , but how to relate the undefined coefficients ...
2
votes
4answers
5k views

Radius of Convergence of power series of Complex Analysis

I have come across the following few questions on past exams papers.. I know how to solve these type when it is of the form $a_nz^n$ but don't have a clue what to do with these. Any help would be ...
2
votes
1answer
206 views

Infinite series involving $\sqrt{n}$

I am looking for examples of infinite series, whose sum is expressed as distributions or known functions, with a $\sqrt{n}$ in each term, such as: $$ \sum_{n=0}^{\infty} \sqrt{n} z^n, \quad ...
11
votes
0answers
367 views

Asymptotic related to the infinite product of sine

The amount is somewhat complicated ($x$ is a constant): $$S_n=\sum_{k=1}^n\ln\left(1-\frac{\sin^2\big(x/(2n+1)\big)}{\sin^2\big(k\pi/(2n+1)\big)}\right)\tag{*}$$ I want to enrich my handy powerful ...
1
vote
2answers
85 views

Sequence convergence of positive numbers

Suppose that $\{a_j\}_j$ is a sequence of real numbers. Suppose for all $j$, $a_j \geq 0$ and the sequence $b_j = \frac{a_j}{1 + a_j}$ converges to $0$. I wish to prove that $a_j$ converges to $0$. ...
2
votes
3answers
121 views

Exercise over sequences of real numbers

Let $(a_n)$ be a sequence of nonnegative real numbers such that $$\lim_{n\rightarrow \infty}a_n=0$$ How to prove that exists an infinite number of indices $ n $ such that $a_n\geq a_m$ for all $m\geq ...
1
vote
2answers
356 views

Geometric series to calculate price

I decided to add my extension to this question as a new question here. I am trying to represent the following as a geometric series equation: ...
1
vote
1answer
117 views

what if geometric sequence + geometric sequence

I wrote a program that basicly can find the formula of the sequence that created with any-degree equation. For example if you give my program that sequence: [1926, 2811, 833240, 28778265, 398155842, ...
5
votes
1answer
268 views

Sum involving the hypergeometric function, power and factorial functions

I am finding some trouble in calculating the following sum involving the hypergeometric function, power and factorial functions. $$ \sum_{y=1}^\infty \mathrm{e}^z \cdot {}_1F_1\left(1-y;2;-z\right) ...
2
votes
1answer
370 views

Difference between convergent sequence and convergent subsequence

I have been thinking about this for a while now. Clearly if a sequence converges then also it will also have a convergent subsequence (take for example the whole sequence). However, I have been told ...
1
vote
1answer
36 views

Prove that for n~=n' sum is much smaller than the case with n=n'

Hi I want to prove that this summation is much smaller for $n\neq n'$ than for the case where $n=n'$. I have seen this fact with simulation results. But I don't know how to prove it in mathematics. ...
3
votes
2answers
150 views

Infinite series question requiring no explanation

Determine if the statement is true or false. No explanation needed. $$\sum_{n=0}^{\infty}\frac{\sin n}{n!}\leq e$$ Although no explanation is needed I was wondering how you would approach this ...
3
votes
3answers
244 views

Approximating an infinite sum of only odd terms by a definite integral

Consider the infinite Sum $S=\sum\limits_{n\ \text{odd}}^{\infty}\left(\dfrac{1}{nt}\right)^2\left[1-i\left(nt\right)^2\right]^{-1}$ Is there a way to approximate this sum as a contour integral? In ...
2
votes
2answers
55 views

Prove the following: Product of Roots

$1^{(1/1)} \cdot 2^{(1/2)} \cdot 3^{(1/3)} \cdot 4^{(1/4)} \cdot 5^{(1/5)} $.... diverges well I don't really know if it does but my gut tells me it does: I can take the log of this product to ...
5
votes
1answer
585 views

Proving that product of two Cauchy sequences is Cauchy

Given that $x_n$ and $y_n$ are Cauchy sequences in $\mathbb{R} $, prove that $x_n y_n$ is Cauchy without the use of the Cauchy theorem stating that Cauchy $\Rightarrow$ convergence. Attempt: Without ...
4
votes
2answers
77 views

Check computation of: $\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}ij$

Compute the below sum: $\sum_{i=1}^{n}\sum_{j=1}^{n}ij$ My working: $\sum_{i=1}^{n}\sum_{j=1}^{n}ij = \sum_{i=1}^{n}i\frac{n(n+1)}{2}$ Now since $\frac{n(n+2)}{2}$ is just a constant we can take ...
4
votes
1answer
459 views

Does $\sum \ln\left(\cos\frac{1}{n}\right)$ converge?

I want to know whether the series $\sum \log\left(\cos\frac{1}{n}\right)$ converges or diverges. I have made some attempts to solve this problem, and I work out them here: $\cos(2\theta) = ...
1
vote
1answer
114 views

Convergence question of Dirichlet's Test

This is an after-chapter exercise of Dirichlet's Test. Show that if the partial sum $S_n$ of the series $\displaystyle\sum_{k=1}^{\infty} a_k \leq Mn^r$, for some $r<1$, then the series ...
1
vote
2answers
59 views

Prove that the series is non-absolutely convergent.

$$a_n = \int_{(2n-2)\pi}^{(2n-1)\pi} \dfrac{\sin t}{t} dt$$ The series is $$\sum_{n=1}^{\infty}a_n$$ I tried using the Cauchy criterion, and this let me with the next inequality: $$\left| S_m - S_n ...
2
votes
1answer
51 views

show the result is right when $f\in C(\Bbb{R})$, but when $f$ is only Riemann integrable. Is it right?

Assume $f(x) \in C(\Bbb{R})$, and $$S_n(x)=\sum_{k=1}^{n}\frac{1}{n}f\left(x+\frac{k}{n}\right),n=1,2,\cdots,$$ show that: $\forall [a,b] \subset \Bbb{R}$ , $S_n$ converges uniformly and if $f(x)$ ...
6
votes
3answers
290 views

What is the half-derivative of zeta at $s=0$ (and how to compute it)?

[Update 3:] I gave a new partial answer following the ansatz in question Q3. I leave the other parts of the question untouched, they are also partially answered in specialized other questions in MSE. ...
0
votes
1answer
79 views

For which $x$ values this series $\sum_{n=1}^{\infty}\frac 1n \cos^2(nx)$ is convergent.

I have tried using Dirichlet's Test with $\cos^2nx = \dfrac{1+\cos(2nx)}{2} = \frac 12 + \dfrac{\cos(2nx)}{2}$, this show that I can't bound the partial sums. What another test may I apply?
1
vote
2answers
72 views

Series, Looks Simple, but I am Stuck

I promised a friend that I could help her about math questions. Yet, I am stuck with a series question. I have written the open form of each term. And I have split the general term into multiples. I ...
1
vote
1answer
169 views

nth term test for divergence - help

$$\sum_{n=1}^{\infty} \left(\dfrac{n}{n+1}\right)^n$$ to show that this diverges should I use the $n^{th}$ term test? So far I have substituted infinity for $n$. Could I use L'hopital's rule to ...
0
votes
1answer
89 views

Wave-Function Series?

So I was basically exploring the function: $\displaystyle {\text{frac}(x)}$ which is the fractional part function and I noticed that it has a nice fourier series definition which is: ...
1
vote
1answer
39 views

question about convergence series

I'm not sure I understand why $$\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}(4pq)^n<\infty $$ when $$p\ne0.5 , p+q=1$$ I know that $\sum_{i=1}^{\infty}\frac{1}{\sqrt{n}}=\infty$ But why when we have ...
1
vote
1answer
60 views

Formula for working out an ID number by given set of coordinates

I'm designing an online game and having a bit of a mental block coding the navigation system. It's designed on a 2 dimensional grid, each cell has an ID 0...n, n being the total number of cells in the ...
1
vote
2answers
71 views

ratio test and divergence

I need to be able to prove that this series converges. I know I need to use the ratio test but I do not know how to go about doing it. Any help is much appreciated! thank you
5
votes
2answers
239 views

Find a closed form for this sequence

$$a_1 = 1; a_2 = 9; a_{n+2} = \frac{a_{n+1}a_n}{6a_n - 9a_{n+1}}$$ I need to find non-recurring formula for $a_n$. Is there any good way to do this? The only one comes to mind is to guess the formula ...