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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A finite sum over $\pm 1$ vectors

$$\mbox{What is a nice way to show}\quad \sum_{\vphantom{\Large A}u\ \in\ \left\{-1,+1\right\}^{N}} \left\vert\,\sum_{i\ =\ 1}^{N}u_{i}\,\right\vert = N{N \choose N/2} \quad\mbox{when}\ N\ \mbox{is even ?.} $$ Could there be a short inductive proof…
Turbo
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Subsubsequence converges $\implies$ sequence converges

Prove that if $\left\{ x_n \right\}$ is an infinite sequence of real numbers, $x \in \mathbb{R}$, and every subsequence $\left\{ x_{n_k} \right\}$ has a subsequence $\left\{ x_{n_{k_j}} \right\}$ with $x_{n_{k_j}} \rightarrow x$, then $x_n…
mr eyeglasses
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How to evaluate the sum

Please help me to evaluate the following $$\sum\limits_{n=1}^\infty \frac{\left(\frac{3-\sqrt{5}}{2}\right)^n}{n^3}$$ Have no idea how to evaluate. Any trick please? Thanks
KON3
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What is the limit of this sequence? It is hard..

Let $a_0 > 0$ $a_{n+1} = \ln(a_n + 1)$ Prove that $\displaystyle\lim_{n\to\infty} n\cdot a_n = 2$
user192276
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Logic behind the multiplicative form of "Gauss trick"?

I was going through $ \lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite I don't follow why/how this is true $(n!)^2 = (1 \cdot n) (2 \cdot (n-1)) (3 \cdot (n-2)) \cdots ((n-2) \cdot 3) ((n-1) \cdot 2) (n \cdot 1)\ge n^n$ I read that At least…
square_one
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Example of a divergent sequence

I want to produce a divergent sequence for which $|x_n - x_{n-1}| \to 0$. So far, I've only been able to show that $$\frac{x_n}{n} \to 0$$, which doesn't really help.
user41281
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Are there other methods to evaluate $\frac{1^{-4}+2^{-4}+3^{-4}+4^{-4}+\cdots}{1^{-4}+3^{-4}+5^{-4}+7^{-4}+\cdots}$?

Are there other methods to evaluate the following series? $$\frac{1^{-4}+2^{-4}+3^{-4}+4^{-4}+\cdots}{1^{-4}+3^{-4}+5^{-4}+7^{-4}+\cdots}$$ My attempt is as…
kiss my armpit
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Mathematical Analysis 2nd ed. - Apostol, Exercise 4.2

If $a_{n+2}=(a_{n+1}+a_n)/2$ for every $n\geq1$, show that $a_n \to (a_1+2a_2)/3$. Now, starting from the fact that $a_{n+2}-a_{n+1}=(a_n-a_{n+1})/2$ and using $b_n=a_{n+1}-a_n$ I obtain $$b_{n+1}=\frac{1}{2}b_n ⇒ b_{n+1}=\frac{1}{2^n}b_1$$ Now I…
Charlie
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Example of a rearrangement that diverges

I know that any conditional convergent series has a rearrangement that diverges. For example if we have $$ \sum_{n=1}^\infty \frac{(-1)^n}{ n} $$ what is a rearrangement that diverges?
Kal S.
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$\lim_{N\to \infty} \sum_{n=N}^{2N} c_n = 0$ $\Rightarrow \sum_{n=1}^{\infty} c_n$ converges?

If $\lim_{N\to \infty} \sum_{n=N}^{2N} c_n = 0$ do we have that $\sum_{n=1}^{\infty} c_n$ converges? At first this did not seem true($\sum_{n=N}^{2N} (-1)^n$ is $0$ when N is odd), but I've failed to find a proper counter-example. I've tried…
iggy
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Find the $n$th term of $1, 2, 5, 10, 13, 26, 29, ...$

How would you find the $n$th term of a sequence like this? $1, 2, 5, 10, 13, 26, 29, ...$ I see the sequence has a group of three terms it repeats: Double first term to get second term, add three to get third term, repeat. What about: $2, 3, 6, 7,…
Guest
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Proving 7n+5 is never a cubic number?

This is from a question that starts with: An arithmetic progression of integers an is one in which $a_n=a_0+nd$, where $a_0$ and $d$ are integers and n takes successive values $0, 1, 2, \cdots$ Prove that if one term in the progression is the cube…
user135842
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Sequence for which no closed form can exist

I was wondering whether there exists a (computable) sequence of numbers, for which it can be proven that no closed form can exist. Edit: By closed form I mean an expression involving only a constant number of elementary functions. So something like…
stefan
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$\sum_{n\geq 0}a_n$ converges iff $\sum_{n\geq 0} \frac{a_n}{\sum_{k=0}^n a_k}$ converges

The last problem I posted had a wrong statement. I recovered the correct one. Let $(a_n)$ be a sequence of positive real numbers. Prove that $\sum_{n\geq 0}a_n$ converges iff $\displaystyle \sum_{n\geq 0} \frac{a_n}{\sum_{k=0}^n a_k}$…
Gabriel Romon
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Proving sequence equality using the binomial theorem

The problem: Prove that for $n \in \mathbb N$: $$ \left(1 + \frac{1}{n} \right)^n = 1 + \sum_{m=1}^{n} \frac{1}{m!} \left(1 - \frac{1}{n} \right) \left(1 - \frac{2}{n} \right) \cdots \left(1 - \frac{m-1}{n} \right). $$ The hint is to use the…
Matt Nashra
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