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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.

4
votes
4answers
173 views

Determine this limit

how can I determine the following limit? $$\lim_{n\to\infty} \frac{\ln\left(\frac{3\pi}{4} + 2n\right)-\ln\left(\frac{\pi}{4}+2n\right)}{\ln(2n+2)-\ln(2n)}.$$ This question stems from this question. ...
1
vote
1answer
12 views

Compact convergence of bivariate complex series

Let $b_n\in H(D(0,R))$ be a sequence of holomorphic functions on a disc of radius $R$ centered at $0$, $\Omega=D(0,R)\times D(0,S)$, $|w_0|<S$. Take a compactly convergent (that is, uniformly ...
2
votes
0answers
22 views

Verify this sum

Apparently this sum has this closed form. $$\sum_{k=j}^{n}{k \choose j}{n \choose k}{n-j \choose k+j}={2j \choose j}{2(n-j)\choose n-j}{n-j \choose 2j}{n+j \choose j}^{-1}$$ How to show that this ...
4
votes
2answers
317 views

Interesting Harmonic Sum $\sum_{k\geq 1}\frac{(-1)^{k-1}}{k^2}H_k^{(2)}$

Here http://integralsandseries.prophpbb.com/topic119.html We came across the following harmonic sum $$\tag{1} \sum_{k\geq 1}\frac{(-1)^{k-1}}{k^2}H_k^{(2)}$$ Note that we define $$H_k^{(2)}=\...
3
votes
5answers
61 views

How to show the sequence $x_n = (1 + \frac{x}{n})^{n}$ is bounded above by $e^x$?

How to show the sequence $x_n = (1 + \frac{x}{n})^{n}$ is bounded above by $e^x$? Note: I'm not supposed to be able to use any differentiation techniques if possible. Since we techincally "don't know"...
0
votes
2answers
32 views

A Proof of the Ratio test connecting it with the Cauchy-Hadamard Theorem

Whilst studying complex analysis I met a proof connecting of the ratio test connecting it with the Cauchy-Hadamard Theorem. Can someone walk me through the proof? I can't seem to understand any of the ...
0
votes
1answer
84 views

The infinite series of $x^{n^2}$

I have some troubles with the following series $$\sum^\infty _{n=0} x^ {n^2}$$ I'm suppose to show that this series is equivalent when $x$ approaches $1$ and $x <1$ to $$\frac{G}{\sqrt{1-x}}$$ ...
1
vote
1answer
26 views

If $f(x) = \sum_{n=0}^{\infty} a_n x^n$ converges for all $x\geq0$, show that $\mathcal{L}\{f\}(s) = \sum_{n=0}^{\infty} \frac{a_n n!}{s^{n+1}}$

If $$f(x) = \sum_{n=0}^{\infty} a_n x^n$$ converges for all $x\geq0$,with $|a_n| \leq \frac{K a^n}{n!}$ for all $n \in \mathbf{N}$ and some constant $K > 0$. I need to show that $$\mathcal{L}\{...
8
votes
1answer
149 views

Proving that $\sum_{k=0}^{2n} {2k \choose k } {2n \choose k}\left( \frac{-1}{2} \right)^k=4^{-n}~{2n \choose n}.$

I have happened to have proved this sum while attempting to prove another summation. Let $$S_n=\sum_{k=0}^{2n} {2k \choose k } {2n \choose k}\left( \frac{-1}{2} \right)^k$$ ${2k \choose k} $ is the ...
0
votes
0answers
14 views

Double harmonic series $\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{H_{n+m}^{(p)}}{(n+1)^{q}(m+1)^{r}}$

Do these sums exist in the literature and have been investigated before? The same question for the odd variant, that is $$ \sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{O_{n+m}^{(p)}}{(2n+1)^{q}(2m+1)^{...
4
votes
2answers
887 views

Bounded sequence in $L^1$ with no weakly convergent subsequence and bounded sequence in $L^\infty$ with no weakly convergent subsequence

Can you give two elementary examples that show: a bounded sequence in $L^1$ with no weakly convergent subsequence; a bounded sequence in $L^\infty$ with no weakly convergent subsequence?? For the $...
0
votes
3answers
32 views

Show that $\left(\frac{n^{\frac{3}{2}}}{2^n}\right)_{n\geq 0}$ is a null sequence. [duplicate]

Show that $\left(\frac{n^{\frac{3}{2}}}{2^n}\right)_{n\geq 0}$ is a null sequence. A null sequence is a sequence tending to $0$. We need to find a $N\in \mathbb{N}$ for every $\varepsilon >0$, ...
0
votes
0answers
35 views

sequence $(x_{n})_{n\geq 1}$ ; $x_{n+1}=x_{n}^2-x_{n}+1$ [duplicate]

I have the following sequence $(x_{n})_{n\geq 1}$ ; $x_{n+1}=x_{n}^2-x_{n}+1$ I need to find $x_1$ such that $(x_{n})_{n\geq 1}$ is convergent. I found that $(x_n-1)^{2}>0$ so $x_n$ is increasing....
8
votes
2answers
152 views
+50

Evaluating sum $\sum_{m=0}^{\infty}\frac{(2-\delta_m^0)(-1)^m \lambda_0}{a(\lambda_0^2 -(\frac{m\pi}{a}))}\cos(m\pi x/a)$

How Can I evaluate the following sum$$\sum_{m=0}^{\infty}\frac{2-\delta_m^0}{a}\frac{(-1)^m \lambda_0}{\lambda_0^2 -(\frac{m\pi}{a})}\cos\left(\frac{m\pi x}{a}\right)=\frac{\cos(\lambda_0 x)}{\sin(\...
27
votes
3answers
705 views

Closed-form of $\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1)=-\int_0^1\frac{\ln(1+x)\ln^3 x}{1-x}\,dx$

Does the following series or integral have a closed-form \begin{equation} \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1)=-\int_0^1\frac{\ln(1+x)\ln^3 x}{1-x}\,dx \end{equation} where $\Psi_3(x)$ ...
-1
votes
0answers
27 views

necessary and sufficient conditions that a number being prime or prime of special form? [on hold]

I like to gather some statements about the properties of prime numbers or prime of the specific forms. For instance 1) A prime number is a whole number greater than 1 whose only factors are 1 and ...
2
votes
2answers
51 views

How can I find the value of this [pathological] function?

A few months ago, while attempting to create a parameterization of the Hilbert curve, I discovered an interesting function, given by the summation... $$f(x)=\sum_{n=1}^\infty \frac{\text{sgn}\left(\...
-1
votes
2answers
42 views

Finding the $n^{th}$ term of unusual sequence

I have this sequence: $7,8,10,13...$. Since this is neither an arithmetic sequence or geometric I was not sure how to go about solving. My initial thoughts are that it goes up by $+1$ then $+2$, then $...
0
votes
0answers
21 views

Polynomials in the Pancake problem

I noticed something interesting in this table. The columns can be expressed by polynomials of order k. I can't check if it is still a polynom for $k=7$. $$k=0: 1$$ $$k=1: n-1$$ $$k=2: n^2-3n+2$$ $$k=3:...
1
vote
1answer
23 views

Evaluation of the series

Given$$ f(x) = \frac{2^x }{2^x +\sqrt{2}}$$ Then find $$S_n= \sum^{2n-1} _{r=1} 2f(\frac{r}{2n})$$ So I tried to evaluate it by adding and subtracting a $√2$ term from numerator , but it didn't help , ...
1
vote
1answer
62 views

Help calculate the limits

Help calculate the limits: 1) $$\lim_{n\to \infty}\sum_{k=1}^n \frac{1}{\sqrt {k(n-k)}}$$ 2) $$\lim_{n\to \infty}\sum_{k=1}^n \frac{1}{(n-k)\ln{n}} $$ 3) $$\lim_{n\to \infty}{n}^p \sin(\pi(\sqrt 2 ...
2
votes
0answers
61 views

Publishing mathematics research(which I believe is already there)

I am a class 12th student, and over the past year have researched quite a lot on sequences and series, and developed a formula to predict : The n-th term of a series The sum to n terms of the series ...
2
votes
1answer
37 views

Exponential Type Series [duplicate]

I'm looking for a closed expression (if it exists) of the following sum: $$\sum_{m=0}^{\infty} \frac{m^n}{m!}c^m$$ where $n \geq 1$ is a positive integer, and $c$ is a fixed constant. The series seems ...
0
votes
0answers
17 views

Ways of calculating Z Transform of a geometric serie

If you have a function $x(n) = 2^nu(n+1)$ And $u(n) = 1 \quad if \quad n > 0$ And you need the Z-Transform you'll have to go through a sum, knowing: $$\sum_{n=0}^\infty 2^nz^{-n} = \frac{1}{1-{...
2
votes
1answer
57 views

Is it possible to obtain the sum of this infinite series? [on hold]

Is it possible to obtain the sum of the infinite series: $$\sum_{n=1}^\infty \frac{c^n}{1-q^n}$$ where $0<c<1, 0<q<1$.
1
vote
1answer
27 views

Couple of questions on Hurwitz theorem

Hurwitz theorem as stated in Hahn and Epstein's Classical Complex Analysis is as follows:
124
votes
22answers
9k views

Why does the series $\sum_{n=1}^\infty\frac1n$ not converge?

Can someone give a simple explanation as to why the harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$ doesn't converge, on the other hand it grows very slowly?...
5
votes
2answers
91 views

Show $ f(x) = \sum_{n=1}^{\infty} \frac{nx}{n^3 + x^3}$ ,$\ g(x) = \sum_{n=1}^{\infty} \frac{x^4n}{(n^3 + x^3)^2}$ are bounded on $[0, \infty)$.

If $f(x), g(x)$ are defined as following on $[0 , \infty)$, $$\tag 1 f(x) = \sum_{n=1}^{\infty} \frac{nx}{n^3 + x^3}$$ $$\tag 2 g(x) = \sum_{n=1}^{\infty} \frac{x^4n}{(n^3 + x^3)^2}.$$ . Then how to ...
1
vote
0answers
22 views

Continued fraction ${0,1,2,3,4,5,6,7,8,9,…}$ and Bessel function [duplicate]

I would like to ask how to get the following relation. \begin{equation} 0 + K_{n = 1}^{\infty} \frac{1}{n} = 0 + \frac{1}{1+ \frac{1}{2 + \frac{1}{3+ \frac{1}{4+ \frac{1}{5 + ...}}}}} = \frac{I_1(...
3
votes
2answers
1k views

An example of a product of two distinct convergent series that is divergent

Do you have an example of a product of two distinct convergent series $\sum x_n$ and $\sum y_n$ $(y_n ≥0)$ that is divergent?
3
votes
2answers
88 views

$a_n $ is a positive integer for any $n\in \mathbb {N} $.

Let $(a_n)_{n\geq 1}$ be a sequence defined by $a_{n+1}=(2n^2+2n+1)a_n-(n^4+1 )a_{n-1} $. $a_1=1$, $a_2=3$. I have to show that $a_n $ is a positive integer for any $n\in \mathbb {N}, n\geq 1$. I ...
1
vote
2answers
38 views

Geometric-like Sum over Primes

Is there a known way to evaluate sums of the form $$\sum_{p\text{ prime}} x^{p},$$ and are there any restrictions on the value of $x$ (e.g., $|x|<1$ for typical geometric series)? EDIT: The ...
0
votes
4answers
73 views

Infinity in infinite series

We define infinite series as summation of terms in sequence. where sequence is defined to function from natural number to real numbers $f:\mathbb{N} \rightarrow \mathbb{R}$. But $\infty$ does not ...
2
votes
1answer
30 views

Algorithm generating subset of primes, can we classify which of them or estimate how large percent of primes are generated?

Assume I have following algorithm: Two lists of numbers, first starting at 2, second starting empty. We now follow rule: Add a number to first list which makes difference with latest number the ...
4
votes
7answers
72 views

Let $x_{n+1} = \frac{1}{2}(x_n + \frac{a}{x_n})$. Prove that $x_{n+1} < x_{n}$

Let $$x_{n+1} = \frac{1}{2}(x_{n} + \frac{a}{x_{n}})$$ Prove that $x_{n+1} < x_{n}$ for $a \geq 0$. Hint: Let the initial guess satisfy $x_{1} > \sqrt{a}$ I am stuck at how to begin this. I ...
1
vote
3answers
78 views

Does the series $\sum_{n=1}^{\infty} \frac{(-1)^n*(n!)^2*4^n}{(2n)!} $ converge?

Does the series $\sum_{n=1}^{\infty} \frac{(-1)^n*(n!)^2*4^n}{(2n)!} $ converge? I have no idea how to do this. I have tried to use any trick I am aware of but can't figure this out. Can anyone help ...
0
votes
2answers
28 views

How to find the limit sum of a series [on hold]

Suppose $$S_n = \lim_{n\to\infty}\frac{\exp(i/2)}{\sum_{j=1}^{i}\exp(j/2)} \ \ \ \text{where} \ \ i = 1,\ldots,n$$ Programatically, $S_n\approx 0.49$ but would I show this by hand?
0
votes
1answer
41 views

Prove by induction that, for all $n\in\Bbb N$, $\sqrt{n} ≤ \sum_ {k=1}^n \frac{1}{\sqrt{k}} < \sqrt{n} + \frac{n}{\sqrt{n+1}}$.

So, I know that for my base case I use $n=1$, and that for the inductive hypothesis we assume the pattern holds until the $n-th$ iteration. Then use that to prove the $(n+1)-th$ iteration ($\Bbb P(n)\...
0
votes
1answer
16 views

Recognizing a Factoring Pattern (Pt. 2)

I am trying to identify a pattern in the following set of equations; $N_{-1}=1$ $N_{0}=2y$ $N_{1}=2y^2+z$ $N_{2}=2y^3+3yz$ $N_{3}=2y^4+5y^2 z+z^2$ $N_{4}=2y^5+7y^3 z+4yz^2$ $N_{5}=2y^6+9y^4 z+...
1
vote
0answers
48 views

Question about big $O$ notation

We all know that exponential functions grow faster than polynomials. Let us consider the following function: $f(n) = n^{a_1} \cdot (\log n)^{a_2}\cdot (\log \log n)^{a_3} \cdot (\log \log \log n)^{a_4}...
0
votes
0answers
56 views

turning $2x$ into a perfect even

So I am trying to generate a sequence with an equation (that I don't think exists) and it involves all the even numbers, and one way to find the sequence is to get rid of all odd prime numbers so... $...
0
votes
1answer
13 views

Is the proportionality characteristic of this function being carried on?

$A=kx$ is a directly proportional function,where $A^2=B^2+C^2$.Does it necessarily mean $B$ and $C$ both vary directly with respect to $x$? If not, under what condition is this possible? Thank you in ...
-1
votes
0answers
34 views

How to express this function as power series? [on hold]

How to express this function as power series $\frac{x}{(2+x^2)^2}$
24
votes
5answers
747 views

Evaluating $\int^1_0 \frac{\log(1+x)\log(1-x) \log(x)}{x}\, \mathrm dx$

In this thread a friend posted the following integral $$I=\int^1_0 \frac{\log(1+x)\log(1-x) \log(x)}{x}\, \mathrm dx$$ The best we could do is expressing it in terms of Euler sums $$I=-\frac{\...
-4
votes
2answers
33 views

Determine the series whether convergence or divergence with using ratio rest. [on hold]

This is the problem: $$\sum_{n=0}^\infty 3^n\sin((\frac{1}{4})^n)$$ I can't prove the convergence of this series, how can we solve it?
0
votes
3answers
41 views

How to prove the double sum of combinations is $3^n$

I have a double sum of combinations as follow $$S = \sum_{i=0}^{n}\sum_{k=i}^{n}{n \choose k}{k \choose i}.$$ I guessed and tested that $S = 3^n$, but I have no idea how to prove this. Any help is ...
1
vote
1answer
109 views

What is the sequence of accumulation points in the 2-adic space, of the Collatz graph?

In the orbit of the function $3x+2^{\nu_2(x)}$ through "accumulation points" of the Collatz graph I have: $?\mapsto\dfrac{-\langle2\rangle\cdot\{5,7\}}{9}\mapsto\dfrac{-\langle2\rangle}{3}\mapsto \...
1
vote
2answers
47 views

Evaluation of series $\sum_{n=0}^\infty\frac{5n+1}{(2n+1)!}$

How to evaluate series $$\sum_{n=0}^\infty\frac{5n+1}{(2n+1)!}$$ I tried to split the summation...but I failed. Please help
0
votes
3answers
76 views

Studying the character of $\sum_{n=3}^\infty \frac{1}{n(\log(\log n))^{\alpha}}$

I have to study the character of this series $$\sum_{n=3}^\infty \frac{1}{n(\log(\log n))^{\alpha}}$$ with $\alpha$ a real parameter. Considering the Cauchy condensation test, the equivalent ...
0
votes
0answers
8 views

Legendre polynomial expansion of a positive function

Let´s assume I have a function $f(\theta)>0$ defined for $\theta<\pi$ and $\theta >0$. I want to find its Legendre polynomial decomposition $f(\theta)= \sum_{l=0}^\infty f_l \, P_l(\cos{(\...