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.

Sequences and series are often considered as a main part of part of calculus, in addition to limits, continuity, differentiation and integration.

A sequence is an enumerated collection of objects in which repetitions are allowed. There are special types of sequences, such as arithmetic sequences (or arithmetic progressions), where the next term is a constant more than the previous; harmonic progressions, which is formed by taking the reciprocal of each term of an arithmetic progression; logarithmic progression, which is formed from a series whose progression getting smaller; and geometric progressions, where the next term is a constant multiplied by the previous term.

A sequence can be given by a direct formula (e.g. $a_n = 2^n + 3$), or by a recurrence relation. In a recurrence relation, the relation between the next term and the earlier terms is given. An example is the recurrence relation $F_{n+2}=F_{n+1}+F_n, n \geq 0.$ Together with the initial terms $F_0=0$ and $F_1=1$, this recurrence relation defines the famous Fibonacci sequence.

A series is formed by summing a sequence. A typical question is: When the number of summed terms goes to infinity, does the sum approach a finite limit? In other words, is it convergent? Several tests, such as the ratio test, the root test, the limit comparison test, the integral test, etc. can help to answer these questions.

Important note. Questions about guessing the next number in a sequence, with no explicit mathematical context, will usually be quickly closed. Consider posting these questions at Puzzling Stack Exchange instead.

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Collatz-ish Olympiad Problem

The following is an Olympiad Competition question, so I expect it to have a pretty solution: For a positive integer $d$, define the sequence: \begin{align} a_0 &= 1\\ a_n &= \begin{cases} \frac{a_{n-1}}{2}&\quad\text{if }a_{n-1}\text{ is…
Raj Raina
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Does an iterated exponential $z^{z^{z^{...}}}$ always have a finite period

Let $z \in \mathbb{C}.$ Let $t = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $a_{n+1} = z^{a_n}$ for $n \geq 1$, that is to say $a_n$ is the sequence $z, z^z, z^{z^z} ...$ This is sometimes called…
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A Ramanujan-type identity: $11\sum_{n=1}^{\infty}\frac{n^3}{e^{2n\pi}-1}-16\sum_{n=1}^{\infty}\frac{n^3}{e^{4n\pi}-1}=\frac{1}{48}$

Out of curiosity, why it is these sums yield a rational answer? $$11\sum_{n=1}^{\infty}\frac{n^3}{e^{2n\pi}-1}-16\sum_{n=1}^{\infty}\frac{n^3}{e^{4n\pi}-1}=\frac{1}{48}$$ I found this identity during the observing ramanujan identiy via a wolfram sum…
user335850
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Asymptotic behavior of iterative sequences

Suppose you have a sequence $$ a_1
Charles
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Does the series $\sum_{n=1}^{\infty}{\frac{\sin^2(\sqrt{n})}{n}}$ converge?

Does the following series converge? $$\sum_{n=1}^{\infty}{\frac{\sin^2(\sqrt{n})}{n}}$$ It shouldn't, but I have no idea how to prove it. I was wondering about Integral Criterion, but the assumptions are not satisfied. Or perhaps Dirichlet test…
joseph
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Conditionally convergent power sums

I'm struggling on the following question: Let $S$ be a (possibly infinite) set of odd positive integers. Prove that there exists a real sequence $(x_n)$ such that, for each positive integer $k$, the series $\sum x_n^k$ converges iff $k \in…
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A sequence with infinitely many radicals: $a_{n}=\sqrt{1+\sqrt{a+\sqrt{a^2+\cdots+\sqrt{a^n}}}}$

Consider the sequence $\{a_{n}\}$, with $n\ge1$ and $a>0$, defined as: $$a_{n}=\sqrt{1+\sqrt{a+\sqrt{a^2+\cdots+\sqrt{a^n}}}}$$ I'm trying to prove here 2 things: a). the sequence is convergent; b). the sequence's limit when n goes to $\infty$. I…
user 1591719
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A problem involving the product $\prod_{k=1}^{n} k^{\mu(k)}$, where $\mu$ denotes the Möbius function

Let $\mu$ denote the Möbius function whereby $$\mu(k) = \begin{cases} 0 & \text{if $k$ has one or more repeated prime factors} \\ 1 & \text{if $k=1$} \\ (-1)^j & \text{if $k$ is a product of $j$ distinct primes}\end{cases}$$…
John M. Campbell
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What is the sum of the reciprocal of all of the factors of a number?

Suppose I have some operation $f(n)$ that is given as $$f(n)=\sum_{k\ge1}\frac1{a_k}$$ Where $a_k$ is the $k$th factor of $n$. For example,…
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Evaluating $f(x) f(x/2) f(x/4) f(x/8) \cdots$

Let $f : \mathbb R \to \mathbb R$ be a given function with $\lvert f(x) \rvert \le 1$ and $f(0) = 1$. Is there a nice simplified expression for $$\begin{align}F(x) &= f(x) f(x/2) f(x/4) f(x/8) \cdots \\ &= \prod_{i=0}^\infty f(x/2^i)?\end{align}$$…
user856
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subset of binary space countable?

I know that the set of all binary sequences is uncountable. Now consider the subset of this set, that whenever a digit is $1$, its next digit must be $0$. Is this set countable? I think it is not because it is like "half" of the set of binary…
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Infinite summation: $x+x+x+x+... =2$?

One of my favourite little math problems is this $x^{x^{x^{x^{...}}}}=2$ The solution to it is quite simple. An infinite tower of x's is equal to 2, and above the first x there is still an infinite tower of x's, so the equation can be simplified to…
user265554
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Average number of times it takes for something to happen given a chance

Given a chance between 0% and 100% of getting something to happen, how would you determine the average amount of tries it will take for that something to happen? I was thinking that $\int_0^\infty \! (1-p)^x \, \mathrm{d} x$ where $p$ is the chance…
小太郎
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The sum of the following infinite series $\frac{4}{20}+\frac{4\cdot 7}{20\cdot 30}+\frac{4\cdot 7\cdot 10}{20\cdot 30 \cdot 40}+\cdots$

The sum of the following infinite series $\displaystyle \frac{4}{20}+\frac{4\cdot 7}{20\cdot 30}+\frac{4\cdot 7\cdot 10}{20\cdot 30 \cdot 40}+\cdots$ $\bf{My\; Try::}$ We can write the given series as $$\left(1+\frac{4}{20}+\frac{4\cdot 7}{20\cdot…
juantheron
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An Inequality for a Trigonometric Sum

Using the equirepartion of the sequence $(n \mod 2\pi)$ one can show that $$\lim_{n\to\infty} \frac1n \sum_{k=0}^n|\cos k|=\frac{2}{\pi}$$ Numerical evidence shows that, for every $n$, $$ \sum_{k=0}^n|\cos k|>\frac{2}{\pi}n.$$ Can someone help…
Omran Kouba
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