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.

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5
votes
2answers
52 views

Regularization of $\sum_{n=2}^\infty (-1)^n \log n$

I accidentally stumbled on the following regularization of this divergent series: $$\sum_{n=2}^\infty (-1)^n \log n "=" \frac{1}{2} \log \frac{\pi}{2}$$ I'm not familiar enough with ...
3
votes
0answers
29 views

“Volterra series” is to “Taylor series” as “Pade approximant” is to _________?

Pade approximants are often better than Taylor series at representing a function. Given a Taylor series, one can use Wynn's epsilon algorithm to easily produce the Pade approximants to it. Volterra ...
0
votes
0answers
24 views

series of independent random variables of descending variance

Let $X_{1},X_{2},\ldots$ be independent random variables such that $\mathbb{E}X_{n}=0$ for all $n$ and $\mathbb{E}X_{n}^{2}=n^{-2}$ for all $n$. Show that the series $\sum_{k=1}^{\infty}X_{k}$ ...
2
votes
2answers
59 views

Prove $\sum_{n=1}^\infty\frac{H_nH_n^{(3)}}{n^2}=\frac{227}{48}\zeta(6)-\frac32\zeta^2(3)$

How to prove this advanced harmonic series of weight 6? $$S=\sum_{n=1}^\infty\frac{H_nH_n^{(3)}}{n^2}=\frac{227}{48}\zeta(6)-\frac32\zeta^2(3)$$ where $H_k^{(p)}=1+\frac1{2^p}+\cdots+\frac1{k^p}$ ...
1
vote
0answers
43 views

Prove verification of “if $\lim \inf a_n=\lim \sup a_n=x$ then $a_n$ converges to $x$”

I know there are proofs of this statement on the website but I was wondering if my, seemingly too simple proof is valid or not. Assume that $\lim \inf a_n=\lim \sup a_n=x$ We know that $\inf_{k\leq ...
0
votes
0answers
38 views

Expression for finding sum of number's digits

In a recent topic I have found this answer (from @Mathlover) that I haven't completely understood, here the link: https://math.stackexchange.com/a/3282980/686644. Someone can explain me how to pass ...
0
votes
0answers
11 views

Recovering primal optimal solutions from dual sub gradient ascent using ergodic primal sequences

My question concerns recovering a primal optimal solution while performing dual sub gradient ascent. Denoting by $y_i$ the dual multiplier in the $i^{th}$ iteration, let \begin{equation} x_i = \...
4
votes
1answer
83 views

Sum of a Infinite sequence

If $$ S = \frac{1}{1\cdot3\cdot5} + \frac{1}{3\cdot 5 \cdot 7} + \frac{1}{5\cdot 7 \cdot 9} \cdots $$ $$S =\, ? $$ My Attempt Let the general term be $ a_n $. Then, $$ a_n = \frac{1}{(2n-1)...
0
votes
4answers
46 views

What does this notation represent? $\lim_{n\to \infty}\frac{1}{n}\sum_{i=1}^{n}a_n$ (Cesaro Means)

My question is as the title, what does this represent?$$\lim_{n\to \infty}\frac{1}{n}\sum_{i=1}^{n}a_i$$ It looks like a series, but it is not as $\frac{1}{n}$ should not be there. Is this just a ...
2
votes
3answers
50 views

Question on connection between series and sequences

I try to understand something I've read: it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as $n$ tends to infinity (if the limit ...
6
votes
1answer
72 views

A series for $\log (a) \log (b)$ in terms of hypergeometric function

I was experimenting with Frullani integral again, and obtained a very curious series: $$\sum_{k=0}^\infty \frac{{_2 F_1} (2k+1,2k+1;4k+2;s)}{(2k+1)^2 \binom{4k+2}{2k+1} r^{2k+1}}= \frac{1}{4} \log (...
6
votes
0answers
69 views

An intriguing sum

While working on the integral posted here, through a large amount of skulduggery, I managed to arrive at the following intriguing sum $$\sum_{n = 1}^\infty \sum_{k = 1}^n \frac{1}{n^4 k 2^k} = 2\...
2
votes
1answer
44 views

Does there exist a family of sequences satisfying the following properties? [on hold]

Does there exist a family $A$ of sequences of distinct terms satisfying the following: If $(x_n),(y_n)\in A$ are distinct, then $\{x_n\},\{y_n\}$ are are disjoint. If $(x_n)\in A,$ then each $x_n$ ...
4
votes
0answers
49 views

A beautiful and efficient series for the logarithm from Frullani integral

Have you seen this series referenced anywhere? $$\sum_{n=0}^\infty \frac{(-1)^n}{1-4 n^2} \binom{2n}{n} \frac{\left(x+ \frac{1}{x}-2 \right)^n}{2^{4n}}= \frac{\sqrt{x}}{2} \left(\frac{\log x}{x-1}...
-1
votes
1answer
48 views

If $\sum_{n=1}^{\infty}a_n$ is absolutely convergent, then $\sum_{n=1}^{\infty}(\frac{n+1}{n})a_n$ is also absolutely convergent? [on hold]

If $\sum_{n=1}^{\infty}a_n$ is absolutely convergent, then $\sum_{n=1}^{\infty}(\frac{n+1}{n})a_n$ is also absolutely convergent. I need converse examples if exists such that above assertion is false....
3
votes
0answers
64 views

Minimum length of sequence such that every integer from 1 to n can be achieved as the sum of some contiguous subsequence

This question literally came to me in a fever dream last night, and it's frustrating me to no end. I'll try to explain it as best I can, but there may not be a satisfying answer; the best outcome ...
0
votes
0answers
24 views

Notation in harmonic series

I was looking at an answer of the post Awfully sophisticated proofs for simple facts, where the topic was the series $\sum_{n=1}^\infty \frac{1}{n} $, and I found this notation: $$ f_n := \frac{1}{n} ...
8
votes
1answer
116 views

$f:[0,1]\to[0,1]$ be a continuous function. Let $x_1\in[0,1]$ and define $x_{n+1}={\sum_{i=1}^n f(x_i)\over n}$.Prove, $\{x_n\}$ is convergent

I have tried a little bit which as follows- Since $f(x_n)\in[0,1]$, $\{f(x_n)\}$ has a convergent subsequence say $y_n=f(x_{r_n})\ \forall n\in\Bbb{N}$ Let, $\lim y_n=l\implies \lim \frac{y_1+y_2+\...
0
votes
0answers
22 views

Limit of summation with values in a range

I'm trying to take the following limit $$\lim_{n\to \infty} \sum_{i=1}^{n} c^{2}_{i} $$where each of the $c_{i}'s$ can take values in $(1/n, 1)$, thus $c_{i}^{2}$ can take values in $(1/n^{2},1)$. My ...
1
vote
0answers
24 views

Does $S=\sum_{k=-\infty}^\infty \frac{(-1)^k\exp ikx}{a^2-k^2}$ converge?

I concluded that $S$ converges ($x\in\mathbb R$, $a\notin\mathbb Z$): $$S=\displaystyle\sum_{k=-\infty}^\infty \dfrac{(-1)^k\exp ikx}{a^2-k^2}=\sum_{k=-\infty}^\infty \frac{(-1)^k \cos kx}{a^2-k^2}+i\...
7
votes
5answers
148 views

Does a general formula for $\sum_{r=1}^{n}r^k$ proof exist? [duplicate]

I know that these are some existing sums that are true: $$\sum_{r=1}^{n}r = \frac{n(n+1)}{2} = \mathcal{O}(n^2)$$ $$\sum_{r=1}^{n}r^2 = \frac{n(n+1)(2n+1)}{6} = \mathcal{O}(n^3)$$ $$\sum_{r=1}^{n}r^3 =...
1
vote
3answers
46 views

Is there a difference infinite sequences and functions?

Is there a difference infinite sequences (all elements are natural numbers) and functions ? I mean for example, Is the infinite sequence $$a_n=\left\{0,1,0,1,0,1 \cdots \right\}$$ equal to $$f(n) ...
-3
votes
0answers
38 views

Proof of $\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}$ [duplicate]

I would like to have a proof of $$\sum_{n=1}^{\infty}\dfrac{1}{n^2} = \dfrac{\pi^2}{6}$$
0
votes
1answer
55 views

Is it possible to define such a set that contains countable many computable and countable many non-computable infinite sequences?

I don't know anything about it. Therefore, I may have used the wrong words in the question. We know that there are uncountable many non-computable infinite sequences which is consist of elements $\...
2
votes
0answers
37 views

Are the reversed digits of $\pi$ equidistributed modulo 1?

Let $r_n$ be the real number obtained by truncating the decimal expansion of $\pi$ at $n$ total digits, removing the decimal point, reversing the digits, then prepending a new decimal point. For ...
2
votes
1answer
60 views

How could I have guess that $-\frac{1}{2}x\sin(x)=\sum_q q (-1)^q \frac{x^{2q}}{(2q)!} $

In some calculations I ended up with the following series: $$\sum_q q (-1)^q \frac{x^{2q}}{(2q)!} $$ I wanted to compute it, I thought it was not really possible, but then I asked mathematica and he ...
2
votes
1answer
84 views

Using integral test to show $ \sum_{n=3}^{\infty} {1 \over {n\cdot \log{n} \cdot \log{\log {n}}}} $ diverge

I want to use the integral test to show that $ \sum_{n=3}^{\infty} {1 \over {n\cdot \log{n} \cdot \log{\log {n}}}} $ diverges. First, I let $ f(x) = {1 \over {x\cdot \log{x} \cdot \log{\log {x}}}} $ ...
-3
votes
2answers
66 views

Solving a set of series [on hold]

What approach would one use to sum the following series? How could the series $$1^2, 2^2, 3^2,\cdots ,n^2$$ and $$k^0, k^1, k^2,\cdots,k^n$$ be solved where $~k~$ is constant? I am not sure exactly ...
3
votes
1answer
62 views

Let $\{f_n\}$ be a sequence of continuous function on a metric space $E$ and $\lim_{n\to \infty} f_n(x_n)=f(x)$ for every $x_n\to x$ and $x\in E $ .

Let $E$ be a metric space where every point of $E$ is an accumulation point. Let $\{f_n\}$ be a sequence of continuous function on $E$ and $$\lim_{n\to \infty} f_n(x_n)=f(x)$$ for every sequence of ...
2
votes
3answers
93 views

Find the interval of convergence of $\sum \frac{x^{2k+1}}{3^{k-1}}$

How do I find the interval of convergence of this series; $$\sum \frac{x^{2k+1}}{3^{k-1}}$$ I have been told that the answer is $$\ -\sqrt{3}<x<\sqrt{3}$$ But I am unsure of where the square ...
4
votes
0answers
70 views

Evaluating series $\sum_{n=2}^\infty \left(n^{1/n}-1-\frac{\log n}{n} \right)$

This is similar to my recent question, but probably more interesting. What can we say about this slowly converging series: $$S=\sum_{n=2}^\infty \left(n^{1/n}-1-\frac{\log n}{n} \right)$$ We can ...
2
votes
2answers
69 views

Two conflicting views on $1_{[n,n+1]}$ [duplicate]

I need to justify whether $(f_{n})_{n}$ as a sequence of real-valued functions converges almost everywhere or not. Note that $f_{n}: \mathbb R \to \mathbb R$ where $f_{n}(x)=1_{[n,n+1]}(x)$. Now I ...
1
vote
1answer
58 views

Find the function of which the given series converges: [on hold]

Find the function of which the given series converges: $$\sum_{n=1}^{\infty} nx^{2n-1} $$ I've tried taking the derivitive (1st and 2nd) and the integral, but all result in divergence. Am I doing ...
1
vote
0answers
23 views

Questions about cyclic numbers, repeating decimals, and full reptend primes

I have a few questions about cyclic numbers in base $b$ ($b = 2$ in particular). We are dealing here with primes $p$ such that the length of the period in the decimal (more precisely base $b$) ...
3
votes
4answers
70 views

How to find Sum of Sum of Partial Harmonic Series: $\sum_{b=1}^{m} \sum_{k=1}^{b} \frac{1}{k} $?

How do I find the Sum of the Sum of Finitely Many Harmonic Series: $\sum_{b=1}^{m} \sum_{k=1}^{b} \frac{1}{k} $? According to maple it is: $\sum_{b=1}^{m} \sum_{k=1}^{b} \frac{1}{k} = \left( \left(...
0
votes
0answers
34 views

Finding the limit of an sequence defined by an integral

I'm looking for a manner to calculate that limit : $$\lim_{j\to\infty}\int^j_{0}re^{-\frac{2j^2}{(j-r)(j+r)}}\frac{(4r^2j^4+4r^4j^2-4j^6+r^2j^8-4r^4j^6+6r^6j^4-4r^8j^2+r^{10})^2}{j^2(j^2-r^2)^8}\,dr$$...
5
votes
2answers
95 views

Is there a sequence $x_n\to+\infty$ such that $\liminf x_{2n}/x_n = 0$?

I have a sequence of real numbers $(x_n)$ that diverges to $+\infty$. Can I conclude somehow that $$\liminf \frac{x_{2n}}{x_n}>0,$$ or are there counterexamples?
-4
votes
0answers
17 views

Is there a way to prove any non-regularity sequence? [on hold]

Of course my answer is not. Because we have a prime number sequence. But If any of you have any information about this, please share it with me.
4
votes
1answer
59 views

Let $f:\mathbb{N} \rightarrow \mathbb{N}$ such that $f(1)=2011$ and $f(n)= \frac{1}{n^2 -1}(f(1)+f(2)+…+f(n-1))$ for $n \ge 2$. Calculate $f(2011)$ [duplicate]

Being $f:\mathbb{N} \rightarrow \mathbb{N}$ such that $f(1)=2011$ and $f(n)= \frac{1}{n^2 -1}(f(1)+f(2)+...+f(n-1))$ for $n \ge 2$. Calculate $f(2011)$ When calculating $f(n)$ I need to consider the ...
2
votes
0answers
22 views

Are there visual proofs for reciprocals of square and triangular numbers?

Similar to this one for the reciprocal of powers of 2 - I couldn't find any. I don't know I used the right terms; maybe it's: infinite series sum. So correct this question if it's wrong.
8
votes
2answers
97 views

Express $\sum_{j=1}^{n}\sum_{i=1}^{n} \frac{1}{i(i+j)}$ in terms of harmonic numbers

Express $$\sum_{j=1}^{n}\sum_{i=1}^{n} \frac{1}{i(i+j)}$$ in terms of the harmonic numbers $H_n$. I guess that there could be several approaches for doing this.
0
votes
2answers
52 views

Construct a sequence with three limit points

Couldn't I just say:$$a_n:=\begin{cases}0, \quad \text{if } n \text{ is even } \\ 1, \quad \text{if } n \text{ is uneven} \\ -1 \quad \text{if } n \text{ is prime}\end {cases}$$ As far as I am ...
1
vote
1answer
42 views

Sum of a sequence. [duplicate]

I would like to find the value of $$\lim_{n\to \infty} \sum_{r=1}^n \frac{r}{1\cdot3\cdot5\cdot7\cdots(2r+1)}.$$ My approach is attached below.
1
vote
1answer
47 views

Limit of the periodic sequence's arithmetic mean

The problem was originally proposed on the national stage of russian national students math olympiad, Sep 2018. Let sequence $x_{n}, n \in \mathbb{N}$, with positive elements - periodic: there ...
1
vote
1answer
69 views

Hardy and Wright irrational sums

A few days ago, I read a proof from Hardy and Wright saying if $a_n$ is an integer sequence and if $$\lim_{n\to \infty}\frac n{a_n}=0$$ then $$\sum_{n=0}^\infty10^{-a_n}$$ is irrational. ...
5
votes
2answers
175 views

High precision evaluation of the series $\sum_{n=3}^\infty (-1)^n (1-n^{1/n})$

This series converges conditionally, but it's quite slow. I would like to find its value with high accuracy: $$S=\sum_{n=3}^\infty (-1)^n (1-n^{1/n})$$ Wolfram Alpha gives $S \approx 0.226354\ldots$....
7
votes
2answers
84 views

Can I get a sequence of bounded functions converging pointwise to $f(x)=1/x$ for $x$ non zero and $0$ for $x$ zero?

How do I construct an explicit sequence of bounded functions converging pointwise to $f(x)=1/x$ for non zero $x$ and $f(0)=0$. It would be better if someone may find a continuous and even better if ...
4
votes
4answers
61 views

Showing $f(x) = \frac{x^2}{\sin(x)}$ is analytic near $0$

Problem Show the function $$ f(x) = \frac{x^2}{\sin(x)} $$ is an analytic about $x=0$. Try We have $$ f(x) = \frac{x^2}{x - x^3/3! + x^5/5! - \cdots } $$ Letting $f(x) = \sum_{n=0}^\...
4
votes
1answer
65 views

How to evaluate $\sum_{i=0}^{j}(-1)^i{j \choose i}(j^n-i^n)^z$

We wish to evaluate this sum $$\sum_{i=0}^{j}(-1)^i{j \choose i}(j^n-i^n)^z$$ where $j=nz$ We can use the binomial to expand $(j^n-i^n)^z$ but it is too messy after. So there must be another way.
6
votes
1answer
121 views

Solve a function based on an inequality

Problem: Suppose that $f(x)\in C^\infty (R),~~$and$~~ \forall n\in N,~~x\in R,~~$we have $$|f^{(n)}(x)|\leq e^x$$Prove that $f(x)=f(0)e^x$. My thought is that prove the derivative of $g=fe^{-x}$ is ...