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
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study the convergence of $\sum_{n=1}^\infty \log \frac{n+1}{n}$

$\sum_{n=1}^\infty \log \frac{n+1}{n}$ $\sum_{n=1}^\infty \log \frac{n+1}{n}$ = $\displaystyle\lim_{n \to{+}\infty}(\log2 - \log1)+(\log3-\log2)+...+(\log(n+1)-\log n)$=$\displaystyle\lim_{n \to{+}\infty}\log(n+1)\to \infty$. So, the series…
Daniela del Carmen
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why this sequence is period $a_{n+5}=a_{n}$

Let $a_{0}=a>0,a_{1}=b>0$,and such $$a_{n+1}a_{n-1}=\max\{(a_{n},1)\},\forall n\in N^{+}$$ show that $$a_{n+5}=a_{n}$$ Even $a,b$ with the 1 uncertainty,so we can't $$a_{2}=\dfrac{\max{(a_{1},1})}{a_{0}}=\begin{cases}\dfrac{a_{1}}{a_{0}}&a_{1}\ge…
user253631
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Value of the series with following terms

A sequence $x_n$ is defined as $x_{k+1}=x_k^2+x_k$ and $x_1=\frac{1}{2}$ Let: $$S_n=\left[\frac{1}{x_1+1}+\frac{1}{x_2+1}+\frac{1}{x_3+1}+\cdots+\frac{1}{x_n+1}\right]$$ where $[\cdot]$ denotes greatest integer function. What is value of…
user247833
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Formula for a sequence

I have this sequence: $-2, 1, 6, 13, 22, 33, ...$ where each term is the previous plus an odd number. The odd numbers are increasing from 3. I am asked to find an explicit formula with respect to $n$ which can give me the $n$-th number in the…
rubik
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Find $\sum\limits_{n=1}^{\infty}\frac{n^4}{4^n}$

So, yes, I could not do anything except observing that in the denominators, there is a geometric progression and in the numerator, $1^4+2^4+3^4+\cdots$. Edit: I don't want the proof of it for divergence or convergence only the sum.
Aditya Agarwal
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Is $\sum_{n=1}^\infty \frac{m}{(n+m)^2}$ bounded for all $m\in\mathbb{N}$?

I'm trying to figure out if there is a finite constant $C$ such that $\sum_{n=1}^\infty \frac{m}{(n+m)^2}\leq C$ for all $m\in\mathbb{N}$. I can see that $\sum_{n=1}^\infty \frac{m}{(n+m)^2}\leq\sum_{n=1}^\infty \frac{m}{n^2}=mc$ for some finite…
JS1204
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Finding an expression for the diameter of a circle.

The figure shows two circles of radius $1$ that touch at $P$. 1]1 $T$ is a common tangent line; $C_1$ is the circle that touches $C$, $D$, and $T$; $C_2$ is the circle that touches $C$, $D$, and $C_1$; $C_3$ is the circle that touches $C$, $D$, and…
user258250
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How to prove that $\sum\limits_{k = 0}^\infty\frac{(- 1)^k}{(2k + 1)^n}$ equals a certain infinite product

I'm trying to see how to get from $$A = 1 - \frac{1}{3^n} + \frac{1}{5^n} - \frac{1}{7^n} + \frac{1}{9^n} - \cdots $$ to $$A = \frac{3^n}{3^n + 1} \cdot \frac{5^n}{5^n - 1} \cdot \frac{7^n}{7^n + 1} \cdot \frac{11^n}{11^n + 1} \cdot…
Pedro
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Why can't this sequence be periodic

Let the sequence $\{a_{n}\}$ be such that $$a_{1}=1,a_{2n}=a_{n},a_{4n-1}=0,a_{4n+1}=1,\forall n\in N^{+}.$$ Show that this sequence can't be periodic. Arguing by contradiction, we assume that there exists a positive integer $T$ such…
user246688
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Convergence of Series w/ factorial

I'm working on a convergence problem that's giving me trouble. I'll list the steps I've made so far. Given the following series determine if it is convergent or divergent: $$\sum_{n=1}^{\infty}\frac{n!\cdot x^n}{n^n}, \text{where } x > 0.$$ When I…
Math_Illiterate
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Summing a binomial series

Consider the following sum: $$S(n)=\sum_{k=0}^{\infty}\frac{\binom{2k+n}{k}}{2k+n}\frac{1}{2^{2k}};n=0,1,2,3,...$$ Is there a closed form for $S(n)$?
Martin Gales
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Radius of convergence of power series or geometric series

To find radius of convergence of geometric series $$\sum_{n=1}^\infty a_n$$ I need to use ratio/root test to find $|L|<1$ To find radius of convergence of power series $$\sum_{n=1}^\infty c_n (x-a)^n$$ I am supposed to find the limit $L$ of just…
Jiew Meng
  • 4,593
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Closed form of power series

First day on the site and this is an amazing place! this is my question: knowing that: $$1+\frac{x^2}{2}+\frac{x^4}{4}+\frac{x^6}{6}+\frac{x^8}{8}+\cdots $$ is a power series, how can I obtain its closed form? Thank you.
Bradzilla
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Finding out $S:=1+\frac12-\frac13-\frac14+\frac15+\frac16-\frac17-\frac18+\cdots$

I was willing to determine the sum of following $$S:=1+\frac12-\frac13-\frac14+\frac15+\frac16-\frac17-\frac18+\cdots$$ I tried the following \begin{align*} S=&\sum\limits_{n=0}^\infty (-1)^n\left(\frac{1}{2n-1}+\frac{1}{2n}\right)\\ …
KON3
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An intriguing relation between two alternating series

This is another intriguing formula coming from Ramanujan's letter to G. H. Hardy dated 16th Jan 1913 $$\frac{\log 1}{\sqrt{1}} - \frac{\log 3}{\sqrt{3}} + \frac{\log 5}{\sqrt{5}} - \cdots = \left(\frac{\pi}{4} - \frac{\gamma}{2} + \frac{\log…
Paramanand Singh
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