Questions tagged [sequences-and-series]
For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.
67,617 questions
-1
votes
0
answers
3
views
Find x in 2, 23, 10, x, 26, 331
2, 23, 10, x, 26, 331
The only help that was given was that the next number in sequence is 50. I've tried with as many AI's possible and every time it yields different answer.
- 1
0
votes
0
answers
29
views
Prove that $4\int_0^1 \frac{\operatorname{Li}_2(x(1-x))}{1-x+x^2}\mathrm{d}x=\int_0^1 \frac{\ln x \ln(1-x)}{1-x+x^2}\mathrm{d}x$
I'd like to prove that
$$4\int_0^1 \frac{\operatorname{Li}_2(x(1-x))}{1-x+x^2}\mathrm{d}x=\int_0^1 \frac{\ln x \ln(1-x)}{1-x+x^2}\mathrm{d}x.$$
Ok, someone said that this holds, but I tried really ...
- 95
1
vote
1
answer
53
views
Possible arrangements for any n number of distinct cubes
This problem has been bouncing around in my head for years, and I can't seem to make progress. I'll give the rules.
Cubes are all uniform in size with an edge length of 1 unit.
Cubes are located ...
- 11
1
vote
1
answer
89
views
True or false? If $\lim n|x_n-x_{n+1}|=0$ then $\{x_n\}$ converges. [duplicate]
I want to know whether it is true that if a real sequence $\{x_n\}_{n=1}^\infty$ satisfies $\lim\limits_{n\to\infty} n|x_n-x_{n+1}|=0$ then it converges.
I guess it is false but I can't find a ...
- 157
1
vote
1
answer
97
views
Find an interval for which $f_n(x) = \frac{x}{1+nx}$ does not converge uniformly
I am studying for my Real Analysis course and one of my practice problems asks us to "prove the sequence of functions $f_n(x) = \frac{x}{1+nx} \to f$ uniformly on certain intervals."
I've ...
-3
votes
0
answers
49
views
Asymptotic equivalence of a binomial Sum :
I've been strugling to find an asymptotic sequence depending on both $\{ap\}$ and $a \in (0,1/2) $ of the following sum defined for all positive integers $p$ :
$$
\sum_{k = 0}^{\lfloor ap \rfloor} \...
-7
votes
1
answer
72
views
Is this constructive method for determining the minimal ε–index of a convergent sequence already known?(Osamah banat theorem for determine least index [closed]
I am studying the ε–definition of convergence for real sequences.
Classical textbooks state that for every , there exists an integer such that
$$|a_n - L| < \varepsilon \quad \text{for all } n \ge ...
-2
votes
0
answers
38
views
Divergence Test [closed]
The divergence test is inconclusive for $f(x) = 1/x$. The sum of the $1/n$'s is in fact divergent, $n\in \mathbb{N}$ and $n$ in $[1, ∞)$. We can say the same for the function $g(x) = 1/x^p$, with $0 &...
- 9
1
vote
0
answers
48
views
Expected number of lattice points in a randomly inscribed triangle
Question. Let $C_R$ be the closed disk of radius $R$ centered at the origin. Let $T$ be a random triangle formed by three vertices $V_1, V_2, V_3$ chosen independently and uniformly from the ...
- 5,309
5
votes
1
answer
123
views
A super-exponential sequence related to the Fibonacci sequence
Consider the sequence $(x_n)_{n\ge 0}$ defined by
$$x_0=1,\;\;\;x_1=1,$$
and for $n\ge 1$
$$x_{n+1}=\sum_{0\le i\lt j\le n}x_ix_j\;\;\;\;\;\;(1)$$
So $x_{n+1}$ is the sum of all pairwise products of ...
- 3,054
2
votes
1
answer
79
views
Prove $B_{2k+1}(1/4) = \frac{-(2k+1) E_{2k}}{4^{2k+1}}$
Question. Is there a simpler way to prove $$B_{2k+1}(1/4) = \frac{-(2k+1) E_{2k}}{4^{2k+1}}$$ where $B_n(x)$ is the $n$-th Bernoulli polynomial and $E_n$ is the $n$-th Euler number?
I have verified ...
- 5,309
17
votes
1
answer
334
views
What is $\lim_n \sqrt[n]{1+\cos(n)}$?
As I was going through some exercise list with limits, I found $\lim_n \sqrt[n]{1+\cos^2(n)}$. This is easy enough, since $\cos^2$ is bounded between 0 and 1, so a squeeze theorem argument lets us ...
- 13.2k
2
votes
0
answers
169
views
Elements belonging to $c_0$ but not to $l^p$
Edit Let $1<p<\infty$. Is there a property $P$ (depending on $p$) for sequences such that for every $(\mu_n)_{𝑛\in\Bbb N}\in c_0$ there are sequences of scalars $(a_n)_{n\in\Bbb N}$ and $(b_n)_{...
- 139
3
votes
1
answer
78
views
$\partial_x f_n(x)=f_{n-1}(x)$ What conditions are needed to guarantee a unique solution?
Consider the functional-recurrence equation
\begin{align}\tag{1}\label{1}
\partial_x f_n(x)=f_{n-1}(x),
\end{align}
what conditions must we impose to guarantee a unique solution?
There are several ...
- 2,506
6
votes
1
answer
175
views
Is $|\sin(n) | \le 1/n^2$ for infinitely many natural numbers $n$?
I have already finished my analysis classes with good grades. But lately I am playing with Geogebra and wonder if there exists a sequence of natural numbers $(n_k)$ such that $$|\sin{n_k}|\leq\frac{1}{...
- 473
0
votes
0
answers
58
views
Swapping sum and integral with an infinite series of modified Bessel functions
I am studying the following integral
\begin{align}
\int_0^{\infty} I_1(\sqrt{x}) \sum_{k=1}^{\infty} \Big( k K_1(k\sqrt{x}) - k^2 \sqrt{x}\, K_0(k\sqrt{x})
\end{align}
where $I_1$ and $K_\nu$ are ...
- 305
4
votes
0
answers
60
views
Reference request for some sums
I have been looking at sums with binomial coefficients in their denominator. These are extensions of Apery's series, which he used in his proof of the irrationality of $\zeta(3)$. This weekend I ...
- 739
1
vote
1
answer
100
views
How to show that the ratio test for $\sum_{n=1}^\infty\frac{(-1)^n\sqrt{n^2+1}}{n\ln n}$ is unity?
According to Wolfram Alpha the ratio test equals $1$ for the series, $\sum_{n=1}^\infty\frac{(-1)^n\sqrt{n^2+1}}{n\ln n}$, hence convergence/divergence is inconclusive. The purpose of this question is ...
- 433
2
votes
6
answers
360
views
Find the limit $\lim\limits_{n\to \infty}\sum\limits_{k=1}^{n}\left(\sqrt{1+\frac{k}{n^2}}-1\right)$
I have this limit
$$\lim_{n\to \infty} \sum_{k=1}^{n} \left(\sqrt{1+\frac{k}{n^2}}-1\right)$$ and and I tried to evaluate it in the following way:
First I set $x=\frac{k}{n^2}$ to make the expression ...
- 359
7
votes
3
answers
257
views
Minimum size of a sequence summing to $2013$ that guarantees a consecutive subset sum of $31$ (still wanted rigorous proof)
I am trying to solve the following problem on integer sequences and subset sums from a 2023 Shanghai high school entrance exam:
Let $A = (a_1, a_2, \dots, a_n)$ be a sequence of positive integers ...
0
votes
0
answers
100
views
Is this method of deriving n!<0 using finite differences valid
From finite differences of $x^n$
I have
$D_nx^n=n!$
which $\implies (D_nx^n)/n!=1$
and telescoping this gives
$S_n=1+1+1+\ldots+1+1+0$, i.e., $= n$ and $D_{n+1}=0$
I have included the integer 1 to ...
- 19
2
votes
1
answer
73
views
if a sequence is in $l^p$
if we have $\sum_{n=1}^\infty |\lambda_n|^q < \infty,$ we may inductively find
$1 = m_1 < m_2 < ...$ such that, defining
$\sigma_k = \{n \in \mathbb{N} : m_k \leq n < m_{k+1}\}$ for $k = 1,...
- 139
3
votes
1
answer
120
views
When is it justified to take a limit inside a series? [closed]
Let
$$
f(x)=\sum_{k=1}^{\infty}(-1)^k(k+1)\,\chi_{\left(\frac1{k+1},\,\frac1k\right]}(x),
\qquad x\in(0,1].
$$
Thus $f$ is constant on each interval $\left(\frac{1}{k+1},\frac{1}{k}\right]$, taking ...
- 537
8
votes
2
answers
232
views
Find $\sum\limits_{i,j,n\ge1}\frac{n + j + i}{n j i (n + j)(n + i)(j + i)}$
how to find the following series:
$$\sum_{i,j,n\ge1}\frac{n + j + i}{n j i (n + j)(n + i)(j + i)}$$
what i attempted was using symmetry like this
\begin{align*}
\sum_{i,j,n \ge 1} \frac{n + j + i}{n j ...
- 83
40
votes
7
answers
2k
views
How far can an infinite number of unit length planks bridge a right-angled gap?
Problem: You are standing at the origin of an infinite flat earth. One quadrant of the earth is gone, leaving an infinite abyss. You have an infinite supply of unit-length zero-width planks. How far ...
2
votes
1
answer
140
views
analytic continuation of the series $\sum_{n=0}^{\infty}\frac{n^2}{\sqrt{a^2+n^2}}$
I would like to understand how to make sense of the following divergent series, or at least to identify the appropriate analytic continuation of its general term:
\begin{align}
\sum_{n=0}^{\infty}\...
- 305
2
votes
1
answer
151
views
Limit with floor sums reminiscent of the exponent of the central binomial coefficient
This problem comes from the 1976 Putnam exam.
Evaluate
$$
L=\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n
\left(
\left\lfloor\frac{2n}{k}\right\rfloor
-2\left\lfloor\frac{n}{k}\right\rfloor
\right),
$$
...
- 23
3
votes
0
answers
97
views
Integer sequence with largest prime factor
Consider the function $F:\mathbb{N}\to\mathbb{N}$ such that $F(n)=\tfrac{n^2-n}{\delta(n^2-n)}$, where $\delta$ returns the biggest prime factor of its input. I wonder if this function always ...
- 965
-3
votes
0
answers
49
views
Divergence of a series based on the sequence [closed]
If a sequence [an] does not converge, that is the limit of "an" as n tends to infinity" does not exist, will the series be referred to as convergent or divergent??
0
votes
0
answers
83
views
Limit of the function satisfying $f(x)=x-f(x^2)$ as $x\to 1^-$
I guess $\lim\limits_{x\to 1^-} f(x) = 1/2$, where the function $f(x)$ defined by $f(x)=x-f(x^2)$ in $[0,1)$, or by the series:
$$
f(x) = x - x^2 + x^4 - x^8 + x^{16} - x^{32} + \cdots.
$$
I know $f(x)...
- 81
0
votes
1
answer
88
views
How many odd numbers are there in one row in Pascal's triangle?
I was looking at the pattern of odd entries in Pascal’s triangle and noticed that every row contains an even count of odd numbers. This is easy to justify, but it led me to wonder how the exact count ...
- 9,329
17
votes
6
answers
1k
views
What is the correct definition of a limit point in real analysis?
This question relates two (seemingly) conflicting definitions of Limit Points in real analysis.
The definition of limit points and closed sets from my notes are written as:
A much more general ...
- 561
3
votes
3
answers
268
views
Analytic sum of an alternating series$\sum\limits_{n=1}^{\infty}(-1)^{n} \frac{n}{\left(n+\sqrt{a+n^2}\right)^2}$
I recently came across the following series with a positive real number $a$:
\begin{align}
S(a) = \sum _{n=1}^{\infty}(-1)^{n} \frac{n}{\left(n+\sqrt{a+n^2}\right)^2}
\end{align}
Does anyone know if ...
- 305
11
votes
3
answers
469
views
Formula for the number of divisors of an integer
I recently derived the following formula for the number of divisors of an integer $n$
$$
D(n)=\lim_{h\longrightarrow 0}h\cdot \pi\cdot \sum_{i=1}^{\infty}\frac{\cot\left( \pi\cdot\frac{n+h}{i} \right)}...
- 333
0
votes
1
answer
112
views
Formal Justification of Alternating Harmonic Series: $1-\frac{1}{2}+\frac{1}{3} - \cdots = \ln(2)$ [duplicate]
It's well known that $\sum_{n=1}^\infty \frac{(-1)^n}{n}=\ln(2)$, which can most easily be seen by the following derivation:
$$\frac{1}{1-x} = \sum_{n=0}^\infty x^n \tag{Geometric Series}$$
$$\frac{1}{...
- 5,547
1
vote
0
answers
71
views
Convergence of Poincaré Series
I am reading Iwaniec’s Topics in Classical Automorphic Forms Chapter 3 on Poincaré Series.
The set up is for some Fuchsian group of first kind $\Gamma$, some multiplier system $\theta$ of weight $k$. ...
- 25
0
votes
2
answers
147
views
Evaluate (when possible) $\sum_{n=0}^{+\infty}\ln\left(2\cos\frac{\alpha}{2^n}-1\right)$
This appeared on the exercises sheet for a "Numerical Series" chapter of a university course: "Determine the nature and the possible sum of the numerical series". Among 18 examples ...
- 197
0
votes
0
answers
87
views
Inequality involving products with harmonic numbers
Let
$$
H_n = \sum_{k=1}^n \frac{1}{k}, \qquad n \ge 1,
$$
and for a fixed parameter $r \in (0,1]$, define a sequence $(a_j)_{j\ge 1}$ by
$$
a_j = 1 - \frac{r}{j}\bigl(H_{j+1}-1\bigr), \qquad j \ge 1.
$...
- 1
7
votes
1
answer
149
views
Does the closest root of a n-th partial sum of a power series to a root of the power series converge?
Let
$$
f(z)=\sum_{k\ge \text{0}} b_{k}\, z^{k}
$$
be a power series with complex coefficients, and suppose $a\in\mathbb {C}$ satisfies $f(a)=0$.
For each index $n$, consider
$$
f_{n}(z)=\sum_{k=\text{...
- 9,329
1
vote
0
answers
54
views
Iterating the map $f(n)=1+D(\sigma(n)-1)$
For integers $n$, the arithmetic derivative $D(n)$ is defined as follows:
$D(p) = 1$, for any prime $p$.
$D(mn) = D(m)n + mD(n)$, for any $m,n\in\mathbb{N}$ (Leibniz rule).
The Leibniz rule implies ...
- 3,054
7
votes
3
answers
316
views
Telescopic series: how to identify it after breaking it down into partial fractions?
I’m trying to understand how to recognize when a series is telescoping. Consider the series
$$
\sum_{n=3}^{\infty} \frac{1}{n(n-1)(n-2)}.
$$
Using partial fraction decomposition, we get
$$
\frac{1}{n(...
- 8,896
12
votes
5
answers
554
views
Non-recursive, explicit rational sequence that converges to $\sqrt{2}$
Is there a non-recursive, explicit sequence of rational numbers that has $\sqrt{2}$ as a limit?
I know of rational sequences such as $x_{n+1}=(x_n+2/x_{n})/2$ and $q_n=[10^n\sqrt{2}]/10^n$ that have $\...
2
votes
1
answer
81
views
Alternative proof to $\sum a_n < \infty \Rightarrow \sum a_n^2 < \infty$ with limit comparison test
If $\sum_{n=1}^{\infty} a_n$ converges and $a_n>0$, then $\sum_{n=1}^{\infty} a_n^2$ converges. There seems to be a standard way to solve this exercise: convergence of $\sum_{n=1}^{\infty} a_n$ ...
- 357
-1
votes
1
answer
102
views
Prove that if $\lim_{n\to\infty}\left(\sum_{i=1}^{n}a_1\cdots a_i^2\cdots a_n\right)$ exists then $\sum a_n,\sum\ln(a_n)$ don't [closed]
Problem: Let $(a_n)_{n=1}^{\infty}$ be a sequence of positive numbers such that
$$\lim_{n\to\infty}\left(\sum_{i=1}^{n}a_1a_2\cdots a_i^2\cdots a_n\right)$$ converges. Then
$$\sum_{n=1}^{\infty}a_n,\...
- 2,262
4
votes
0
answers
160
views
Defining an integral over surreal numbers
Im not an expert on the surreals but I have noticed that even Conway when writing the 2nd edition of his book mentioned that a natural definition of an integral over surreals is still elusive. So I ...
- 1,747
3
votes
1
answer
75
views
Counting unique-loop configurations on a $3 \times n$ grid
This is a smaller post that relates to these previous questions that I have asked $(1)$ $(2)$. Context for the (seemingly arbitrary) formulas may be found there.
Let us consider a $3 \times n$ grid ...
- 5,309
-1
votes
1
answer
116
views
Is there an explanation for the recurring sequence of final digits in the series of differences between cubes? [closed]
In the sequence of differences between the cubes of successive integers, 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397 and so on, the sequence of their final digits repeats: 1, 7, 9, 7, 1; 1, 7, ...
- 117
1
vote
1
answer
132
views
Divergence, convergence of the series $\sum_{n=1}^{\infty} \left[ 2^{(n^x - n^x \cos(1/n^2))} - 1 \right]$
I have this series
$$\sum_{n=1}^{\infty} \left[2^{\left(n^{x}-n^{x} \cos \frac{1}{n^{2}}\right)}-1\right].$$
let $a_n$ the general term of the series,
$$
a_n = 2^{\left(n^x - n^x \cos \frac{1}{n^2}\...
- 8,896
2
votes
0
answers
111
views
Subsequential limits of $(\cos(k))_{k\in \mathbb{N}}$
I am looking at a text book which is a reference in my educational background, and I usually find it to be a reliable source, but I am struggling with one of the proofs in it:
We'd like to prove that ...
- 177
2
votes
1
answer
121
views
Minimum number of forbidden cells for a unique loop on $m \times n$ grid
I recently asked about the minimum number of black (forbidden) squares needed to guarantee a unique "loop" cycle on an $n \times n$ grid. It seems natural, then, to consider $m \times n$ ...
- 5,309