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
5
votes
3 answers

how to prove that $(1 + \frac{1}{n})^{n+1}$ is decreasing?

Please, help me to prove that $$x_n=\left(1+\frac{1}{n}\right)^{n+1}$$decreases. I know I must to prove that that $$\frac{x_n}{x_{n+1}}> 1$$ What to do next?
Walter r
  • 1,097
5
votes
2 answers

Periodic sequence problem

Given sequence $a_n$ defined such that $a_1=3$, $a_{n+1}=\begin{cases}\frac{a_n}{2},\quad 2\mid a_n\\ \frac{a_n+1983}{2},\quad 2\nmid a_n\end{cases}$. Then prove that the sequence $a_n$ is periodic and find the period. It's easy to prove that…
Mutse
  • 571
5
votes
1 answer

Summing series containing $e^{an} \pm 1$ term in the denominator

I was looking over Ramanujan's first letter to Hardy and came across several series of a similar form: $$ \sum_{n=1}^{\infty} \frac{n^{13}}{e^{2\pi n}-1} = \frac{1}{24} $$ $$ \sum_{n=1}^{\infty} \frac{\coth(n\pi)}{n^7} = \frac{19 \pi^7}{56700} $$ $$…
Dave
  • 1,723
5
votes
2 answers

Convergent Sequence Terminology

What is the following sequence classified as? I don't want to make anybody solve it, I just need to know where to begin looking to solve it. $$\alpha_1 = \sqrt{20}$$ $$\alpha_{n+1} = \sqrt{20 + \alpha_n}$$ I am suppose to prove that it converges to…
Addo Solutions
  • 225
5
votes
1 answer

Monotonicity in alternating Series

For the convergence of an alternating series, the sequence $\{p_n\}$ needs to be a non-negative, monotonically decreasing sequence with a limit of zero. However, I'm having difficulty thinking of an example where the absence of monotonicity is an…
user71468
  • 53
5
votes
2 answers

Why does the series $\sum\limits_{n=2}^\infty\frac{\cos(n\pi/3)}{n}$ converge?

Why does this series $$\sum\limits_{n=2}^\infty\frac{\cos(n\pi/3)}{n}$$ converge? Can't you use a limit comparison with $1/n$?
Billy Thompson
  • 1,860
  • 7
  • 23
  • 23
5
votes
1 answer

Is there another name for this "power of two" sequence?

I was recently asked how to call a number sequence, and since I am not sure about naming conventions, I am grateful for any help. There is a sequence of real numbers, where: the next element always equals the previous element, multiplied by 2 1 is…
ISE
  • 153
5
votes
1 answer

Calculus 2: Series Integration

I do not understand how$\int \sum_{n=0}^\infty (-1)^nx^{2n}dx = \sum_{n=0}^\infty (\frac{(-1)^nx^{2n+1}}{2n+1})+C$. Here are my steps: $$\int \sum_{n=0}^\infty (-1)^nx^{2n}dx$$ $$=\sum_{n=0}^\infty (-1)^n\int x^{2n}dx$$ $$=\sum_{n=0}^\infty …
user532874
5
votes
1 answer

Determining the value of $\sum_{n=2}^\infty \frac{1}{n^n-1}$

$$\displaystyle\sum_{n=1}^\infty\sum_{m=2}^\infty \frac{1}{m^{mn}}=\sum_{n=1}^\infty\left(\frac{1}{2^{2n}}+\frac{1}{3^{3n}}+\frac{1}{4^{4n}}+\cdots\right) \tag{$\star$}$$ I have a strong suspicion that the above summation converges, although I'm not…
C. Melton
  • 338
5
votes
4 answers

Study the convergence of $x_n$, $ x_{n+1}=\frac{1}{2}\big(\frac{x_n+3}{ x_n}\big)$, with $x_0=1.$

How can we prove that the limit of the sequence $x_n$ defined by: $ x_{n+1}=\frac{1}{2}\big(\frac{x_n+3}{ x_n}\big)$, with $x_0=1$ exists? I have tried to prove that it is Cauchy , but I failed? Can I get some help, and thanks in advance.
Kamal
  • 648
5
votes
1 answer

Smallest possible sum $\sum_{i=0}^{\infty} \frac{a_i^2}{a_{i+1}}$ over non-increasing infinite sequences starting with 1

Let $a_n$ be a sequence of positive numbers such that $a_0=1$ and $\forall n : a_n \ge a_{n+1}$. Find the infinum of $\sum_{i=0}^{\infty} \frac{a_i^2}{a_{i+1}}$ over all such sequences. If $\{a_n\}$ is a geometric seriess $1, q, q^2, ...$, where…
kvardekkvar
  • 2,196
5
votes
0 answers

Is it possible to give an elementary evaluation for this sum: $ \sum_{n=0}^{\infty}(x^{(2^{-n})}-1)$

I've been thinking about this sum for a little while. I came up with it based on the observation that the product: $$\prod_{n=0}^{\infty}x^{(2^{-n})}$$ converges to $x^2$, and if we take the natural log of this product then we…
Robo300
  • 1,248
  • 4
  • 9
5
votes
1 answer

Convergence test of a trigonometic series

Test the convergences of the following series $$ \sum_{n=1}^{\infty} (-1)^n \frac{ \sin n}{\sqrt{n}}$$
math123
  • 379
  • 1
  • 10
5
votes
1 answer

Cauchy sequence and sums of distances of consecutive terms

Consider a sequence $(x_n)$ in metric space $(X,d)$. If the sequence of partial sums $\sum_{i=1}^n d(x_i,x_{i+1})$ of distances between consecutive terms converges in $\mathbb{R}$, then I understand why the sequence is Cauchy. But is the converse…
Mark
  • 53
5
votes
0 answers

Can the result of $\sum\limits_{n=1}^{\infty}\dfrac{1}{3^n-2^n}$ be expressed as a closed form?

Notice that, for any $a>b>0$ and $n=1,2,\cdots$, it holds that \begin{align*} a^n-b^n&=(a-b)(a^{n-1}+a^{n-2}b+\cdots+ab^{n-2}+b^{n-1})\\ &\geq…
mengdie1982
  • 13,840
  • 1
  • 14
  • 39
1 2 3
…
99 100