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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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Find a general form of a sequence and its sum

I have a problem to find a general form of the sequence \begin{align} - \frac{{n\left( {n - 1} \right)}}{{2\left( {2n - 1} \right)}},\frac{{n\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right)}}{{2 \cdot 4 \cdot \left( {2n - 1}…
Mohammad W. Alomari
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What is the nth term of the lucky numbers?

If a sequence is generated like so: start with the odd numbers $1,3,5,7,...$ start at $n=1$ take the next number $n$ from the sequence. So at the start of the first pass: sequence is $1,3,5,7,...$ and $n = 3$ remove every $n\mathrm{th}$ number from…
minseong
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Find all $a$ such that $\{x_n\}$ has finite limit

Let $\{x_n\}$ be a real sequence defined by: $$ x_1=a \\ x_{n+1}=3x_n^3-7x_n^2+5x_n $$ For all $n=1,2,3...$ and $a$ is a real number. Find all $a$ such that $\{x_n\}$ has finite limit when $n\to +\infty$ and find the finite limit in that cases. My…
tangkhaihanh
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Generalization of Leibniz formula for $\pi$

The well-known Leibniz formula for $\pi$ is $$ 1-\frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \ldots = \frac{\pi}{4}. $$ Looking at some nonstandard techniques, I've happened upon the following formula: \begin{equation}…
user387394
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Limit of series

I'm sure this has been asked a million times, but it's hard to google for a particular series without knowing its name. $$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$$ I know this converges absolutely to $\frac{\pi^2}{6}$ and I know that it is absolutely…
countunique
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$\sum _{n=1}^{\infty} \frac 1 {n^2} =\frac {\pi ^2}{6}$ and $ S_i =\sum _{n=1}^{\infty} \frac{i} {(36n^2-1)^i}$ . Find $S_1 + S_2 $

I know to find sum of series using method of difference. I tried sum of write the term as (6n-1)(6n+1). i don't know how to proceed further.
raj
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"Uniform convergence" of $\frac{k(k+1)\cdots(k+n-1) - k^n}{n!k^n}$

Does the expression $$\phi(n,k)=\frac{k(k+1)\cdots(k+n-1) - k^n}{n!k^n}$$ uniformly converge to zero as $k\to \infty$? more precisely, given $\varepsilon>0$ can we find $N$ such that for all $k\geq N$ and all $n$ it holds that $\phi(n,k)<…
Daniel
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Product of two infinite summations

I want to show \begin{equation*} \sum_{n=1}^\infty f(n)\sum_{m=1}^\infty g(m)=\sum_{n=1}^\infty \sum_{m=1}^\infty f(n)g(m)\end{equation*} Do I have to invoke the conditions on Fubini's theorem? Or is this always true? I tried this proof, which does…
telemaco
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Is this proof that $\sum_{k=1}^n \frac{1}{k}$ is never an integer a dead end?

I am familiar with the proof using Bertrand's Postulate as well as the proofs highlighted in this discussion: Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer? However, I wanted to try my hand at proving…
user373763
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Does $\sum\sqrt{n\arctan(1/n^3)}$ converge?

Does the following series converge? Why? $$\sum\sqrt{n\arctan(1/n^3)}$$ The only way that comes to me is to look at the series: $$\arctan(x)=\sum\frac{(-1)^k}{2k+1}x^{2k+1}\Longrightarrow…
3x89g2
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Prove that a sequence converges to $\sqrt{2}$

Given $x_1 = 2$ and $x_{n+1} = \dfrac{1}{2} x_n + \dfrac{1}{x_n}$, prove that $x_n \to \sqrt{2}$. I thought I could use monotone convergence but I have a hard time proving the monotonicity of the sequence.
Lundborg
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Convergence of the series $\sum_{n=1}^{\infty }\frac{1}{n}\left ( \arctan n-2\arctan\frac{n-1}{n} \right )$

Does the series: $\sum_{n=1}^{\infty }\frac{1}{n}\left ( \arctan n-2\arctan\frac{n-1}{n} \right )$ converge or diverge? I've tried to apply Taylor expansion, but I wasn't able to find a proper way to apply it.
Phuong Vu
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Find $\lim\limits_{n\to+\infty}(u_n\sqrt{n})$

Let ${u_n}$ be a sequence defined by $u_o=a \in [0,2), u_n=\frac{u_{n-1}^2-1}{n} $ for all $n \in \mathbb N^*$ Find $\lim\limits_{n\to+\infty}{(u_n\sqrt{n})}$ I try with Cesaro, find $\lim\limits_{n\to+\infty}(\frac{1}{u_n^2}-\frac{1}{u_{n-1}^2})$…
tangkhaihanh
  • 917
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If $\sum\frac{a_n}{b_n}$ and $\sum\frac{a_n^2}{b_n^2}$ both converge then $\sum\frac{a_n}{a_n + b_n}$ converges

If $\sum\limits_{n=0}^\infty\frac{a_n}{b_n}$ converges and $\sum\limits_{n=0}^\infty\frac{a_n^2}{b_n^2}$ converges, where $(a_n + b_n)b_n \ne 0$ for every $n \in \mathbb{N}$ , then show that $\sum\limits_{n=0}^\infty \frac{a_n}{a_n + b_n}$ also…
Ester
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When can we switch the order of $\liminf$ and have inequality?

Here is my question. Let $a_{m,n}$ be a positive sequence, and I have that $a_{m,n}\leq L<+\infty$ for all $n$ and $m$. I also know that $\lim_{m\to \infty}a_{m,n}= a_n$ for each $n$ fixed. I understand that if I have $$ \lim_{m\to…
spatially
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