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

Proving that $\sin(n)$ is not uniformly distributed modulo $1$

How do you prove that $\sin(n)$ with $n=0,1,2...$ is not uniformly distributed mod 1? (This is an exercise in Uniform Distribution of Sequences by Kuipers and Niederreiter.)
user7084
4
votes
3 answers

Boundedness of $a+\frac 1a$ when iterated

Here's something I was wondering... Is $$a + \frac 1a$$ for any positive real number $a$ bounded when iterated? For example,, if we start at $a=1$, continuing gives us $a= 1+ \frac 11=2$, then $a=2+\frac 12=2.5$ and so on. A quick program shows…
Macaroni
  • 43
4
votes
5 answers

Evaluate the sum of series $\sum_{k=1}^{+\infty}(-1)^{k-1}\frac{1}{k(4k^2-1)}$

Evaluate the sum of series $$\sum_{k=1}^{+\infty}(-1)^{k-1}\frac{1}{k(4k^2-1)}$$ I have tried two methods: 1) using power series 2) using partial sums but I can't find the sum. 1) Using power…
user300045
  • 3,449
4
votes
1 answer

Find the sum of the infinite series: $ 1 + \frac{1+2}{2!} + \frac{1+2+2^2}{3!} +\frac{1+2+2^2+2^3}{4!}+...$

Find the sum of the infinite series $$ 1 + \frac{1+2}{2!} + \frac{1+2+2^2}{3!} +\frac{1+2+2^2+2^3}{4!}+... ....$$ What I have done let $$ S = \underbrace{\frac{1}{1!}}_{\text{1st Term}} + \underbrace{\frac{1+2}{2!}}_{\text{2nd Term}} + …
bigfocalchord
  • 5,597
4
votes
1 answer

Sum of a nearly classic series

Assuming we know that : $$\sum_{n=1}^{+\infty}{\frac{1}{n^2}} = \frac{\pi^2}{6}$$ How do you find the sum of a series where all terms are in this one ? For instance, how do you prove that ?$$\sum_{n=1}^{+\infty}{\frac{1}{(2n-1)^2}} =…
Cydonia7
  • 891
4
votes
2 answers

Find the General coefficient in the MacLaurin Series $(1+x)^{(1+x)^{(1+x)^{...}}}$

The first nine terms of the MacLaurin series of the following function is: $$(1+x)^{(1+x)^{(1+x)^{...}}}= 1+x+x^2+\frac{3}{2}x^3+\frac{7}{3}x^4+4x^5+\frac{283}{40}x^6+\frac{4681}{360}x^7+\frac{123101}{5040}x^8+...$$ This can be verified by…
Linus Choy
  • 245
  • 1
  • 7
4
votes
0 answers

When does $\sum_{n\in\mathbb{Z}}(a_n+b_n)=\sum_{n\in\mathbb{Z}}a_n+\sum_{n\in\mathbb{Z}}b_n$?

$a_n,b_n\in\mathbb{C}$. Is it enough that both $\sum_{n\in\mathbb{Z}}a_n$ and $\sum_{n\in\mathbb{Z}}b_n$ converge? It seems to me that this should be enough,…
Tom83B
  • 670
4
votes
2 answers

The sum of series $1-1/2-1/4+1/3-1/6-1/8+1/5-....$?

I know that the sum of series $1-1/2+1/3-1/4+1/5..= log 2.$ And we can see that by rearranging the terms of the series given in question, i would land on the series as above. so this should mean that the answer should be log2. but no, the answer is…
Parul
  • 783
4
votes
3 answers

Show that $\sum\limits_{n=1}^{32}\frac1{n^2}=1 + \frac{1}{4} +\frac{1}{9} +\dots+ \frac{1}{1024}<2$

Show that $$1 + \frac{1}{4} +\frac{1}{9} +\dots+ \frac{1}{1024} <2$$ I know that the denominators are perfect squares starting from that of $1$ till $32$. Also I know about this identity $$\frac{1}{n(n+1)} > \frac{1}{(n+1)^2} >…
Yami Kanashi
  • 703
  • 3
  • 11
4
votes
1 answer

Help find closed form:$\sum_{n=1}^{\infty}{m!\over n\cdot n(n+1)(n+2)\cdots(n+m)}$

show that and determine the closed form $$\sum_{n=1}^{\infty}{m!\over n\cdot n(n+1)(n+2)\cdots(n+m)}=\sum_{n=1}^{\infty}{{1\over n} }\sum_{k=0}^{m}(-1)^k{m\choose k}{1\over (n+k)}$$ $m\ge0$ My try: I was observing the favourite Euler's…
gymbvghjkgkjkhgfkl
  • 8,090
4
votes
3 answers

Does the series $\sum\limits_n(2^{1/n} - 1)$ converge?

Does the series $\sum\limits_{n=1}^\infty (2^{1/n} - 1)=(2^1 - 1) + (2^{\frac{1}{2}} - 1)+ ... +(2^{\frac{1}{n}}-1)+...$ converge? I feel like it diverges the same way as $\sum \frac{1}{n}$ diverges: very slow, on a logarithmic scale.
Saketh Malyala
  • 13,637
4
votes
6 answers

The digits of a positive integer, having three digits, are in A.P and their sum is $15$. The number ..

The digits of a positive integer, having three digits, are in A.P and their sum is $15$. The number obtained by reversing the digits is $594$ less than the original number. Find the original number. My Attempt: Let the three digits number be…
pi-π
  • 7,416
4
votes
3 answers

How can we evaluate this $\prod_{k=1}^n(1+kx)$

$\displaystyle\prod_{k=1}^n(1+kx)=\underbrace{\displaystyle\sum_{k=0}^n a_k x^k}_{\text{I assumed this,it don't have to be like this}}$ I'm investigating what this means, how we can analyse this and get generalized formula. In fact ,I thought…
Micheal Brain Hurts
  • 2,380
4
votes
2 answers

Summation manipulation.

How can this be true? \begin{align*} \sum_{n=1}^\infty \sum_{k=n}^\infty \frac 1{k^3} &= \sum_{k=1}^\infty \sum_{n=1}^{k} \frac 1{k^3}\\ \end{align*}
whoisitnow
  • 131
4
votes
3 answers

$\forall n \in \mathbb{N^*}$, prove that $|{\frac{\sqrt{n^2+2}}{2n} - \frac{1}{2}}| < \frac{1}{2n^2}$

Let $$x_n = \frac{\sqrt{n^2+2}}{2n},n \in \mathbb{N^*} \ldots$$ Prove that $$|{x_n - \frac{1}{2}}| < \frac{1}{2n^2}$$ Indication: Use the relationship: $\sqrt{1+\gamma} < 1 + \frac{\gamma}{2}$, $ \forall \gamma \in \mathbb{R^{*}_{+}}$ I'd appreciate…
user42268
1 2 3
…
99 100