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
4
votes
1 answer

Finding Limit of Expression involving Recursive Sequence

Let $(x_n)_{n\geq1}$ be a sequence such that $x_1 = x >1$ and \begin{align} x_{n+1}=x_n + \sqrt{x_n} - 1, n\geq 1\\ \end{align} Evaluate \begin{align} \lim_{n\to \infty} \frac{4x_n -n^2}{n \log n} \end{align} My attempt: Let $y_n=\sqrt{x_n}$, we…
Rishi
  • 833
4
votes
3 answers

Sequence which changes its sign like this

Which is the general formula of this sequence? $$ x_0 = -1$$ $$ x_{n+1} = ((-1)^n*X_n)/2^n$$ What baffles me more is the sign which is like this: $--++--++--++--++\cdots$ I've been wondering how the sign could change like that, any ideas?
Gtoyos
  • 53
4
votes
2 answers

Baby Rudin Chapter 3 Exercise 11(d)

I'm writing solutions to exercises in Baby Rudin I think might want to assign students, and I'm having particular difficulty with 11(d) in Chapter 3. Namely, for a sequence $(a_{n})_{n=1}^{\infty}$ of positive reals such that…
Matt Rosenzweig
  • 4,201
4
votes
3 answers

Computing $\sum\limits_{n=2}^{\infty }\frac{1}{n^3-n}$

I don't understand why I can't get the telescopic sum after the partial fraction decomposition: $$\frac{1}{n^3-n}= \frac{1}{n(n-1)(n+1)}=\frac{-1}{n}+\frac{1}{2(n-1)}+\frac{1}{2(n+1)}=\frac{-1}{n}+\frac{n}{(n-1)(n+1)}.$$ I have…
mezzaluna
  • 319
  • 1
  • 5
4
votes
1 answer

Prove that $a_n>100$ when $n>10$ (linear recurrent sequence)

Consider the sequence defined by $a_0=a_1=1$ and $a_{n}=2a_{n-1}-3a_{n-2}$. Here, a proof is given that $|a_n|\to +\infty $. I would like to prove that $|a_n|>100$ when $n>10$. How can we do it ? (Also, is there an explicit lower bound for the…
Friedrich
  • 1,312
4
votes
2 answers

Determine for which $\alpha > 0$ the series $\sum_\limits{n=1}^{\infty}\frac{ne^n-\log(1+n)}{n}(\frac{1}{n^{\alpha}})$ converges

Determine for which $\alpha > 0$ the following series converges $$\sum_\limits{n=1}^{\infty}\frac{ne^n-\log(1+n)}{n}\frac{1}{n^{\alpha}}$$ My…
Lorenzo B.
  • 2,252
4
votes
1 answer

Why radius of convergence is $\frac{1}{r}=\liminf_{n\to \infty }\left|\frac{a_{-n-1}}{a_{-n}}\right| ?$

Let consider the series $$\sum_{n\in\mathbb Z}a_nz^n.$$ We denote $R$ the radius of $\sum_{n=0}^\infty a_nz^n$ and $r$ the radius of $\sum_{n=-\infty }^{-1}a_nz^n$, i.e.e the series converge absolutely if $r<|z|
user330587
  • 1,624
4
votes
1 answer

$\sum_1^\infty{\frac{1}{n(n+1)(n+2)}}$?

How to find the sum of the series $\sum_1^\infty{\frac{1}{n(n+1)(n+2)}}$? I expanded it via partial fractions but it does not look like a telescoping series which I was expecting. Am I missing something obvious or easy manipulation here?
blabla
  • 1,104
4
votes
6 answers

Creating a series that skips every $n^\text{th}$ term?

I want to write a function using sigma notation that could represent an arbitrary number of terms of, for example, $1+2+4+5+7+8+10+11+13\ldots$, skipping every third term. I think one would need functions like floor and mod, but I'm not certain.
Jonathan Doliver
  • 41
4
votes
1 answer

Continuity of a series of functions at zero

Let $\mathrm{sinc}(x) = \sin(x)/x$ if $x\neq 0$ and $\mathrm{sinc}(0) = 1$. This is a smooth function. Let $(a_n)$ a real sequence such that $\sum_n a_n$ converge, and let $$ f(x) = \sum_{n=0}^{+\infty} a_n \,\mathrm{sinc}(nx)^2. $$ I am trying to…
Yann
  • 474
4
votes
1 answer

Characterizing subsequences of the Thue-Morse sequence

Consider the Thue-Morse sequence on the alphabet $\{0,1\}$ given by $T_0 = 0$ and $T_{n+1} = T_n \bar{T_n}$ where $\bar{T_n}$ is the bitwise negation of $T_n$. Then the Thue-Morse sequence is defined as $$TM:=\lim\limits_{n\to\infty}T_n$$ (this is…
Slugger
  • 5,556
4
votes
6 answers

Calculate the sum of $\sum_{n=1}^{\infty} \frac{x^{n+1}}{n(n+1)}$

I would like to calculate the sum for $x\in(-1,1)$ of this: $$\sum_{n=1}^{\infty} \frac{x^{n+1}}{n(n+1)}$$ So far I managed that $$\int \frac{x^n}{n}dx = \frac{x^{n+1}}{n(n+1)}, $$ and $$\sum_{n=1}^\infty \frac{x^n}{n}=-\log(1-x), $$ and $$ \int…
Juan Carlos
  • 169
  • 7
4
votes
5 answers

Tribonacci Sequence Term

A tribonacci sequence is a sequence of numbers such that each term from the fourth onward is the sum of the previous three terms. The first three terms in a tribonacci sequence are called its seeds For example, if the three seeds of a tribonacci…
bio
  • 638
4
votes
1 answer

Derivative of $ f(x) = \sum\limits^{\infty}_{n=0} \frac{4n-7}{6n+7} x^n$

Consider $$f(x) = \sum^{\infty}_{n=0} \frac{4n-7}{6n+7} x^n.$$ Find $ f'(x).$ I simply took the derivative which I thought is $$\sum^{\infty}_{n=0} \frac{4n^2-7n}{6n+7} x^{n-1}.$$ The response says: "This is a very subtle mistake. You have included…
Haim
  • 757
4
votes
2 answers

Suppose that $\sum a_n$ converges and that $a_n \neq 0$ for all $n \in \mathbb N $. Prove that $\sum \frac{1}{a_n}$ diverges.

Suppose that $\sum a_n$ converges and that $a_n \neq 0$ for all $n \in \mathbb N $. Prove that $\sum \frac{1}{a_n}$ diverges. My attempt let the series $\sum a_n$ converge to a. Therefore $\lim_{n\to\infty} S_n = a$ Yeah. I have never tackled a…
Tinler
  • 1,061
1 2 3
…
99 100