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.
67,600 questions
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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
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What is the correct definition of a limit point in real analysis?
This question relates to two (seemingly) conflicting definitions of Limit Points in real analysis.
The definition of limit points and closed sets from my notes is written as:
A much more general ...
- 459
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1
answer
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Formula for the number of divisors of an integer
I recently derived the following formula for the number of divisors of an integer $n$
$$
D(n)=\lim_{h\longrightarrow 0}h\cdot \pi\cdot \sum_{i=1}^{\infty}\frac{\cot\left( \pi\cdot\frac{n+h}{i} \right)}...
- 155
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Convergence of Poincaré Series
I am reading Iwaniec’s Topics in Classical Automorphic Forms Chapter 3 on Poincaré Series.
The set up is for some Fuchsian group of first kind $\Gamma$, some multiplier system $\theta$ of weight $k$. ...
- 25
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57
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Inequality involving products with harmonic numbers
Let
$$
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
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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 ...
- 2,992
7
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3
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278
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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
$$
\frac{1}{n(...
- 8,886
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5
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Non-recursive, explicit rational sequence that converges to $\sqrt{2}$
Is there a non-recursive, explicit sequence of rational numbers that has $\sqrt{2}$ as a limit?
I know of rational sequences such as $x_{n+1}=(x_n+2/x_{n})/2$ and $q_n=[10^n\sqrt{2}]/10^n$ that have $\...
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Alternative proof to $\sum a_n < \infty \Rightarrow \sum a_n^2 < \infty$ with limit comparison test
If $\sum_{n=1}^{\infty} a_n$ converges and $a_n>0$, then $\sum_{n=1}^{\infty} a_n^2$ converges. There seems to be a standard way to solve this exercise: convergence of $\sum_{n=1}^{\infty} a_n$ ...
- 357
4
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Defining an integral over surreal numbers
Im not an expert on the surreals but I have noticed that even Conway when writing the 2nd edition of his book mentioned that a natural definition of an integral over surreals is still elusive. So I ...
- 1,737
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Counting unique-loop configurations on a $3 \times n$ grid
This is a smaller post that relates to these previous questions that I have asked $(1)$ $(2)$. Context for the (seemingly arbitrary) formulas may be found there.
Let us consider a $3 \times n$ grid ...
- 5,205
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Divergence, convergence of the series $\sum_{n=1}^{\infty} \left[ 2^{(n^x - n^x \cos(1/n^2))} - 1 \right]$
I have this series
$$\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}\...
- 8,886
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answer
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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 $m \times n$ ...
- 5,205
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Sequence on OEIS [closed]
A friend and I the sort of found the sequence A381587 at https://oeis.org/A381587 by ourselves, and found that is was already on OEIS. Does it really contain no even numbers greater than 2?
- 129
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55
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Help with asymptotics of a summation
Fix $s_0 > 1$. Suppose $s_1 \in (s_0, s_{\max}]$, where $s_{\max} > s_0$ is
arbitrary. Let $K = [s_0, s_{\max}]$. For $n \in \mathbb{N}$, define $D_n : K \times K \rightarrow [0,
\infty)$ by
\...
- 12.9k
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Is it necessary to prove that convergent sequences have unique limits? [closed]
I am learning sequences and series. Whenever I try to read rigorous formulations of this topic, I encounter a proof for uniqueness of limit of a convergent sequence. I wonder why is it necessary to ...
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What are the properties of the "Taylor series for numbers"? [closed]
In this answer mentioned the factorial number system, which allows to represent any rational number in $[0,1)$ as a sum of inverse factorials:
$0.123_! = \frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!} = \...
- 10.6k
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double sum $\sum _{m=1}^{\infty}\sum_{\substack{n=1 \\ n \neq m}}^{\infty}\frac{1-\left(-1\right)^{n+m}}{n^2-m^2}$
I am trying to compute the double sum
$$S = \sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1-(-1)^{n+m}}{n^2-m^2}.$$
Note that $1-(-1)^{n+m} = 0$ if $n+m$ is even and $2$ if $n+m$ is odd, so the sum is ...
- 51
3
votes
1
answer
188
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Iterating the arithmetic-derivative map $U(n)=n+D(n)−1$
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).
$D(-n) = -D(n)$.
The ...
- 2,992
18
votes
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answers
320
views
A sequence based on arithmetic derivative that always converges to prime numbers
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).
$D(-n) = -D(n)$.
The ...
- 2,992
16
votes
1
answer
532
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$\int_0^{\infty}\text{Ai}^4(x)dx = \ln(3)/24 \pi^2$
How can I prove that
$$\Omega = \int_{0}^{\infty} \text{Ai}^4(x) \, dx = \frac{\ln(3)}{24 \pi^2}$$
where $\text{Ai}(x)$ is the Airy-function.
Using the Fourier integral representation of the Airy ...
- 5,205
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What conditions on the sequence $a_i$ is it true that $n$ is finite?
Define $(a_i)_{i = 1}^{\infty}$ to be a sequence of positive integers and assume that $n$ denotes the number of positive integers not the sum of numbers in the sequence $(a_i)_{i = 1}^{\infty}$. What ...
- 109
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How to extend Erdős problem $409$ to $\mathbb{N}^3$?
Erdős problem 409 ("How many iterations of $n↦\phi(n)+1$ are needed before a prime is reached? Can infinitely many $n$ reach the same prime? What is the density of $n$ which reach any fixed prime?...
- 2,992
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Extended Discussion on Catalan Number
As we all know if $$C_n = \sum_{i=0}^{n-1} C_{n-i-1}\:C_{i},\quad C_1 = C_0 = 1; \tag{1}$$
then the general formula is $\frac{1}{n+1}\dbinom{2n}{n}; $
I solve it by using combinatorics. However, if ...
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Is it true that every sequence converging to zero has a subsequence that is eventually completely monotonic?
In searching for a special structure within any sequence converging to zero, specifically a structure stronger than monotonicity, I propose the following construction:
First, consider the definition ...
- 11