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.

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Does the sequence $\sin(n!\pi^2)$ converge or diverge?

Does the sequence $\sin(n!\pi^2)$ converge or diverge?
M.R. Yegan
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Determine if sum of series is rational or not

Is there some methods to find out if the sum of an infinite series is rational or not if we have no closed form expression for the sum? For instance: $$\begin{align*} &\sum_{n=1}^{\infty} \frac{n^2}{n!+1}\\ &\sum_{n=1}^{\infty} \frac{1}{n^{7/2}…
grok_it
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Convergence of $\frac{\sqrt{a_{n}}}{n}$

Can anyone help me with the following question. If $a_{n} \geq 0$ and $\sum a_{n}$ converges then how to prove $\sum \frac{\sqrt{a_{n}}}{n}$ converges. Any idea where to start. My idea was to try using comparison test since $\sqrt{a_{n}} \leq…
jpc
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Find $\sum_{n=0}^\infty\frac{2^n}{3^{2^{n-1}}+1}$

Find $\sum_{n=0}^\infty\frac{2^n}{3^{2^{n-1}}+1}$. I already proved that it converges but I can't find the sum, Although I think it should be $1+\frac{1}{\sqrt3 +1}$, from calculations.
Plom
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How to prove that this sequence converges? ($a_n=a_{a_{n-1}}+a_{n-a_{n-1}}$)

I have some problems, with the convergence of this sequence defined recursively. It's clear that it's bounded. But is it convergent? How can I check for convergence? $$ a_0 = a_1 = 1 $$ $$ a_n = a_{a_{n - 1} } + a_{n - a_{n - 1} } $$ How do I…
Daniel
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Convergence of an infinite series

Does the series $\displaystyle\sum_{n \geq 1} \frac{1}{n^{2 + \sin n}}$ converge? Why? It seems as if this series will converge since $2 + \sin n > 1$ for all integers $n$, but since $2 + \sin n$ is arbitrarily close to $1$ for some $n$ I cannot…
user93184
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How fast does the sequence $y_t$ defined by $y_{t+1}=y_t(1-y_t)$ decay to zero?

The question is in the title; I'm looking for the exact decay rate. Naturally, assume the starting point $y_0$ belongs to $(0,1)$. This is motivated by one of the answers to a previous question of mine.
robinson
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General Term for the Sequence 0, 1, 5, 6, 13, 14, 18, 19, 29, 30, 34, 35, 42

I am interested in the following sequence which I came upon while generalizing an Integral…
Miracle Invoker
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$x_1=1$, $x_2=2$, $x_{n+2}=\frac{x_{n+1}+x_n}{x_{n+1}-x_n}$ is divergent?

In my analysis class they let us find out if the following sequence converges or not: $$x_1=1,\quad x_2=2,\quad x_{n+2}=\frac{x_{n+1}+x_n}{x_{n+1}-x_n}$$ This sequence appears in the Bartle but the exercise does not ask to prove convergence. The…
Luis Alexandher
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Prove that for every $n\in \mathbb{N}^{+}$, there exist a unique $x_{n}\in[\frac{2}{3},1]$ such that $f_{n}(x_{n})=0$

Let $f_{n}(x)=-1+x+\dfrac{x^2}{2^2}+\dfrac{x^3}{3^2}+\cdots+\dfrac{x^n}{n^2}$, (1) Prove that for every $n\in \mathbb{N}^{+}$, then there exist unique $x_{n}\in[\frac{2}{3},1]$ such that $f_{n}(x_{n})=0$ (2) Show that the sequence $(x_{n})$ of (1)…
math110
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Is every Wolstenholme number greater than or equal to its index?

Background This question was inspired by this code golf post, and I've taken some of this background explanation from there. Consider the generalised harmonic numbers of order 2: $$H_{n,2} = \sum^n_{k=1} \frac 1 {k^2}$$ This sequence begins: $$1,…
Jonah
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Prove that a recurrent sequence $u_n$ is convergent if $\lim \limits_{n \to \infty}(u_{n+1}-u_n)=0$

Let $f:[a;b] \to [a;b]$ be a continuous function ($a$ and $b$ real numbers) and $(u_n)$ a sequence defined by $u_0 \in [a;b]$ and $u_{n+1}=f(u_n)$ Prove that if $\lim \limits_{n \to \infty}\left(u_{n+1}-u_n\right)=0$ then $(u_n)$ converges I tried…
Alex
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Given the sequence $a_1=1$,$a_{n+1}=1+\frac{n}{a_n}$, does the sequence $a_n+n-a_n^2$ converges?

Given the sequence $a_1=1$,$a_{n+1}=1+\frac{n}{a_n}$, does the sequence $a_n+n-a_n^2$ converges? If it converges, find the limit. Now, I proved $\frac{1+\sqrt{1+4(n-1)}}{2}\leq a_n\leq \frac{1+\sqrt{1+4n}}{2}$, and $0\leq p_n=a_n+n-a_n^2\leq…
M.Grape
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Insight about $\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{\cos(nx)\cos(my)}{n^2+m^2}$

Can someone give me some insight about the following double sum? I would be deeply appreciated. $$\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{\cos(nx)\cos(my)}{n^2+m^2},$$ where $x,y\in[-\pi,\pi]$. I don't even know if it converges for…
PML
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Calculate $\frac13+\frac29+\frac{1}{27}+\frac{2}{81}+\frac{1}{243}+\frac{2}{729}+\ldots$

How can we calculate the formula for the following series? $\frac13+\frac29+\frac{1}{27}+\frac{2}{81}+\frac{1}{243}+\frac{2}{729}+\ldots$ I'm not quite sure what type of series it is anyway. It doesn't look geometric. Is there any strategy/procedure…
Bob Shannon
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