Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

1
vote
0answers
13 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 ...
3
votes
5answers
56 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"...
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}\{...
0
votes
2answers
31 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
0answers
13 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)^{...
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....
0
votes
3answers
31 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 $\varepsilon >0$, ...
-1
votes
2answers
41 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 , ...
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 $$\frac{G}{\sqrt{1-x}}$$ ...
2
votes
0answers
60 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 ...
3
votes
4answers
151 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. ...
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
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(\...
2
votes
1answer
35 views

Exponential Type Series

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 ...
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(...
-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
1answer
54 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
2answers
37 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 ...
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 ...
0
votes
2answers
27 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
40 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+...
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
votes
0answers
34 views

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

How to express this function as power series $\frac{x}{(2+x^2)^2}$
-4
votes
2answers
32 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
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... $...
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
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}...
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:
0
votes
3answers
40 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 ...
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 ...
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{(\...
3
votes
1answer
47 views

Convergence/Divergence speed of $u_n = \int_0^1 f(x)^n g(x) \textrm{d}x$ given $f, g$ continuous and non-negative

Let be $f, g : [0, 1] \to \mathbb{R}_{+}^{*}$ continuous maps such that: $\forall n \in \mathbb{N}, u_n = \int_0^1 f(x)^n g(x) \textrm{d}x$ I want to show that $v = \left(\dfrac{u_{n + 1}}{u_n}\...
0
votes
1answer
48 views

For $a>0$,the series $\sum_{n=1}^\infty a^{\ln n}$ [duplicate]

For $a>0$,the series $\sum_{n=1}^\infty a^{\ln n}$ is convergent if and only if (1). $0<a<e$ (2). $0<a\leq e$ (3). $0<a<\frac{1}{e}$ (4). $0<a\leq \frac{1}{e}$ I tried ...
0
votes
5answers
34 views

Sequence divergence test

Could I argue that $\left(1+\frac{1}{n}\right)^{n^2}$ = $e ^n$ therefore the sequence diverges? I am wondering if it is a legal move
2
votes
2answers
38 views

Why does $\sum_{n=1}^\infty \frac{1}{n(\log(n))^{1+2\epsilon}}$ converge?

I am looking through examples on convergences of random series, and in one of the proofs the following result is used: If $\epsilon > 0$ then $$\sum_{n=1}^\infty \frac{1}{n(\log(n))^{1+2\epsilon}}&...
0
votes
1answer
43 views

Recognising a Factoring Pattern

I am trying to identify a pattern in the following set of equations; $$N_0=y$$ $$N_1=y^2+z$$ $$N_2=y^3+2yz$$ $$N_3=y^4+3y^2z+z^2$$ $$N_4=y^5+4y^3z+3yz^2$$ $$N_5=y^6+5y^4z+6y^2 z^2+z^3$$ Essentially, ...
0
votes
1answer
18 views

Is there a special name for numbers whose multiples remain multiples when reversed?

I recently noticed that reversing multiples of 11 gives you other multiples of 11, for example 11 x 19 = 209 & 902 ÷ 11 = 82 ...is there a name for that? &, if so, does that same name ...
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
1answer
33 views

Struggling with finding a potential counterexample for a convergent series.

This question comes with two parts. Part (a): Let $\{f_n(x)\}$ be a sequence of nonnegative functions for $x \in S \subseteq \mathbb{R}$ such that $f_1 \geq f_2 \geq \dots \geq 0$, and that $f_n \to ...
-1
votes
0answers
64 views

Let $f_n$ is continuous function from $[0,1]$ to $\mathbb{R}$ for any natural $n$. And for any $x$ from $[0, 1]$ series $\sum_n f_n(x)$ converge. [on hold]

Prove that exist a positive length interval $[a, b]$ in $[0, 1]$ such that partial sums of series $f_n$ is evently limited in $[a, b].$
3
votes
3answers
86 views

Suppose $x_n$ is a decreasing sequence of positive reals with $\sum x_n$ converges, must $(n\log n)x_n \to 0$

We are able to show, using the Cauchy criterion (using sum from $n$ to $2n$) that $nx_n \to 0$ Explicitly this is $0<nx_{2n}<\displaystyle \sum_{i=n}^{2n}x_n$ and the result follows from squeeze ...
0
votes
1answer
32 views

Convergence of a sequence… [duplicate]

Let $\{a_n\}$ be a sequence of real numbers. Define $\sigma_n = 1/n(a_1 + \dots + a_n)$. Suppose that $\lim a_n = a \in \mathbb{R}$. Show that $\lim \sigma_n = a$. Here is my work so far... Fix $\...
1
vote
1answer
61 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 ...
3
votes
1answer
111 views

Let $x_{n} = \sqrt{1 +\sqrt{2 + \sqrt{3 + \dots \sqrt{n}}}}$. Show $\lim_{n \rightarrow \infty} x_{n}$ exists. [duplicate]

Let $x_{n} = \sqrt{1 +\sqrt{2 + \sqrt{3 + \dots \sqrt{n}}}}$. show $\lim_{n \rightarrow \infty} x_{n}$ exists. To do this the problem has been broken down into three pieces: a) Show that $x_{n} <...
2
votes
1answer
55 views

Theorem 30.1 (b) in Munkres' TOPOLOGY, 2nd ed: The sequential criterion for continuity

Here is Theorem 30.2 in the book Topology by James R. Munkres, 2nd edition: Let $X$ be a topological space. (a) Let $A$ be a subset of $X$. If there is a sequence of points of $A$ converging ...