Mathematics Stack Exchange
Mathematics Stack Exchange

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.

65378 questions
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Can this sum ever converge?

If I have a strictly increasing sequence of positive integers, $n_1
user56914
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Integer absolute difference sequences

Given an arbitrary sequence of nonnegative integers $x_0, \ldots, x_{n-1}$, define its difference sequence as $x'_i = | x_i - x_{i+1} |$, with indices mod $n$ (so $x_n = x_0$). Repeated application of this process leads to interesting behavior. For…
Joseph O'Rourke
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How to find this $\prod_{k=-\infty}^{+\infty}\frac{x^2+(4k+1-y)^2}{x^2+(4k+3-y)^2}$

let $x,y$ be real numbers,show that $$\prod_{k=-\infty}^{+\infty}\dfrac{x^2+(4k+1-y)^2}{x^2+(4k+3-y)^2}=\dfrac{1+e^{\pi x}-2e^{\frac{\pi}{2}x}\sin{\dfrac{\pi}{2}y}}{1+e^{\pi x}+2e^{\frac{\pi}{2}x}\sin{\dfrac{\pi}{2}y}}$$ This is a question that a…
math110
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Value of $ \sum \limits_{k=1}^{81} \frac{1}{\sqrt{k} + \sqrt{k+1}} = \frac{1}{\sqrt{1} + \sqrt{2}} + \cdots + \frac{1}{\sqrt{80} + \sqrt{81}} $?

I tried my best, but I am totally clueless about it. Worse thing is we were supposed to arrive at the answer in approximately $ 2 $ minutes. The correct answer is $ 8 $, right? Can you kindly explain how to arrive at it? I hope it won’t be too much…
Ivy Mike
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What is the asymptotic behavior of A103213 in OEIS?

It's probably not at all hard—but at least right now it's not obvious to me—how to determine the asymptotic behavior of $\sum_{k=1}^n \binom{n}{k} \frac{1}{k}$ (link to OEIS).
S Huntsman
  • 465
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formula for $\sum_{n = 1}^{\infty}{\frac{1}{a^n+b^n}}$

Is there some sort of formula for the infinte sum: \begin{equation} \sum_{n = 1}^{\infty}{\frac{1}{a^n+b^n}} \end{equation} (one can assume $a,b>1$) What I've got so far: $$\sum_{n = 1}^{\infty}{\frac{1}{a^n+b^n}}\le\sum_{n =…
Sagi Shadur
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What does "definition is independent of the choice of" mean?

What does "definition is independent of the choice of" mean? An example: Let $W$ be Banach and $V$ a normed space. Let $X$ be a dense subspace of $V$. Let $T \in Lin(X,W)$. For every $v \in V$ there exists $(x_k) \in X$ s.t. $\lim_{k \rightarrow…
mavavilj
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Example of Convergent Series

Can anyone think of sequences $\{a_n\}$, $\{b_n\}$ such that $\sum a_n$ diverges, ${b_n}\to\infty$, but $\sum a_nb_n $ converges? Thank you. Note that $\{a_n\}$ must have infinitely many positive terms and infinitely many negative terms. Edit: I get…
Daniel
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What would be the general form of $\sum_{i=1}^n \frac{1}{i}$?

How would I come to a general form of this type of summation, similar to $$\sum_{i=1}^n i = \frac{n(n+1)}{2}$$ and $$\sum_{i=1}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$ ? I found this post about calculating higher powers of i, but am unsure if this…
Jonah
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Closed form for $\sum_{k=0}^\infty \frac{x^k}{(k+1)^4 k!}$

A graduate student in physics asked me if there was a closed form for the following series, which he had found as the second virial coefficient for a gas in an exponential potential: $$\sum_{k=0}^\infty \frac{x^k}{(k+1)^4 k!}$$ where $x < 0$. This…
Connor Harris
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Given $x_0=\sqrt{2}+\sqrt{3}+\sqrt{6}$ and $x_{n+1}=\frac{x_n^2-5}{2(x_n+2)}$, how can I find $x_n$?

Let $x_0=\sqrt{2}+\sqrt{3}+\sqrt{6}$ and $$\forall n \in \mathbb{N} : x_{n+1}=\dfrac{x_n^2-5}{2(x_n+2)}$$ Then $x_n=?$ My Try : $$x_{n+1}(2x_n+4)=x_n^2-5 \\2x_{n+1}x_n+4x_{n+1}=x_n^2-5\\x_n^2-2x_{n+1}x_n-(4x_{n+1}+5)=0$$ So we have…
Almot1960
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Prove that $\sum\limits_{n=1}^\infty\frac{\text{Si}^2(\pi n)}{n^2}=\frac{\pi^2}2$

I was doing numerical calculations and found $$\sum _{n=1}^{\infty } \left(\frac{\text{Si}(\pi n)}{\pi n}\right)^2\overset{?}{=}\frac{1}{2},$$ where $\text{Si}(x)$ means the sine integral. Interestingly, it seems that only when the terms are…
Aster
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How to show that $\sum_{n=1}^{\infty}{2^n\over (2^n-1)(2^{n+1}-1)(2^{n+2}-1)}={1\over 9}?$

How to show that $${2\over (2-1)(2^2-1)(2^3-1)}+{2^2\over (2^2-1)(2^3-1)(2^4-1)}+{2^3\over (2^3-1)(2^4-1)(2^5-1)}+\cdots={1\over 9}?\tag1$$ We may rewrite $(1)$ as $$\sum_{n=1}^{\infty}{2^n\over (2^n-1)(2^{n+1}-1)(2^{n+2}-1)}={1\over…
gymbvghjkgkjkhgfkl
  • 8,090
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Computing $ \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}\frac{1}{\Gamma \left(\frac{m+n}{2}\right)\Gamma \left(\frac{1+m+n}{2} \right)}$

I'm trying to compute: $$ \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}\frac{1}{\Gamma \left(\frac{m+n}{2}\right)\Gamma \left(\frac{1+m+n}{2} \right)}$$ (From CMJ) Using the duplication formula: $$ \Gamma(x)\Gamma \left(x+\frac{1}{2}…
Chon
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How does this series scale ? (A fractional Touchard polynomial)

I was wondering how does series $$ \sum_{k=1}^{\infty} \frac{x^k}{k!} \sqrt{k} $$ scale with $x \rightarrow \infty$. And I am trying to prove that whether the limit $$ \lim_{x \rightarrow \infty} \frac{\sum_{k=1}^{\infty} \frac{x^k}{k!}…
Liang
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