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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Show that the series $\sum\limits_{n\ge1}(n+1)^nn^{-n-3/2}$ converges

Consider $\sum_{1}^{\infty} a_n$ , where $a_n$ =$\dfrac{(n+1)^n}{n^{n+\frac{3}{2}}}$ we find $$\dfrac{a_n}{a_{n+1}}=[\dfrac{(n+1)^2}{n(n+2)}]^n\times \dfrac{(n+1)^{\frac{5}{2}}}{(n+2)n^{\frac{3}{2}}}$$ by D-Alembert ratio test , $\lim_{n\rightarrow…
user120386
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Investigating $\sum_{n=1}^\infty \frac{\log{n}}{n^c}$

I want to make sure my solution is correct, also possibly see some other solution from the people of Math.SE $$\sum_{n=1}^\infty \frac{\log{n}}{n^c},c\in\mathbb{R}$$ For $c\leq 0$ series clearly diverge, as the sequence does not converge to 0. For…
Dahn
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How find this minimum of the $a$ such $x_{n}>0$

let real numbers $a>0$, and sequence $\{x_{n}\}$ such $$x_{1}=1,ax_{n}=x_{1}+x_{2}+\cdots+x_{n+1}$$ Find the minimum of the $a$,such $x_{n}>0,\forall n\ge 1$ My try:…
user94270
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Sum of Series with negative exponents

$ 3= \sum_{n=1}^{t} \frac{1}{1.08^n} $ I see that it is $3 = 1.08^{-t}(12.5 \times 1.08^t{-12.5})$ (from Wolfram Alpha, but I'm not sure how to get it. I tried solving as a geometric series, I had problems and didn't get the correct answer. I see…
s c
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Find general term of a sequence

What would be the best way in finding a general term $a_{n}$, $n \geq 2$ for the recursive sequence $a_{n} = 3a_{n - 1} + 1$, where $a_{1} = 1$.
user2850514
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Double summation for this series

$$ \sum_{n=0}^{\infty}\left(\frac{x^n}{n!}\sum_{r=0}^{n-1}\left(\frac{y^r}{r!}\right)\right) $$ This is a double summation I need to evaluate(not for a homework problem, but for a probability question involving gaming I found). I can't find any idea…
The Wayne
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Computing the nth term of a sequence when n is really large

How to find the, say, 28383rd term of the sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2,.... ? EDIT: The sequence is the sequence of digits of positive integers in order. thanks,
yati sagade
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Series $\sum\limits_{n=1}^{\infty}\frac{n^k}{n!}$

Is there a formula for $\displaystyle\sum_{n=1}^{\infty}\frac{n^k}{n!}$ in terms of $k$? What about $\displaystyle\sum_{n=1}^{\infty}\frac{(n-1)^k}{n!}$?
user85798
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Summability vs Unconditional Convergence

I'm reading that summability and unconditional convergence are the same. (At least if the index set is countable, so unconditional convergence makes sense.) Then unconditional convergence is nothing else than convergence of the net of all rearranged…
C-star-W-star
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Series used in proof of irrationality of $ e^y $ for every rational $ y $

Following from the book "An Introduction to the Theory of Numbers" - Hardy & Wright I am having trouble with this proof. The book uses a familiar proof for the irrationality of e and continues into some generalizations that lose me. In the…
Hemofelicity
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Find $\sum ^{\infty }_{n=2}\dfrac {2+4+6+\ldots +2n}{1+a_{n}}$

Define $a_{n}=1!\left( 1^{1}+1+1\right) +2!\left( 2^{2}+2+1\right) +3!\left( 3^{2}+3+1\right) +\ldots +n!\left( n^{2}+n+1\right) $ , Find $\sum ^{\infty }_{n=2}\dfrac {2+4+6+\ldots +2n}{1+a_{n}}$
user109004
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Squared fractional sum inequality

Let $n$ be a positive integer. Is it true that $$\sum_{k=n+1}^\infty\dfrac{1}{k^2}<\dfrac1n?$$
JJ Beck
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Multivariate generating function

I am investigating the perturbation of the Jordan canonical form. In my work I must calculate the number of ways to factor $p^{(n-k)} q^k$ where $p$ and $q$ are distinct primes (https://oeis.org/A054225). This sequence is generated by the function:…
Alexander
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What is the value of $ a_{2009}$

I have the following sequence : $a_0 = 3 $ $ a_n = 2 + a_0 a_1 a_2\text{ ...}a_{n-1} $ How can I find the value of $a_{2009} $ ?
Way to infinity
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Let $\{a_n\}$ be a positive monotonic decreasing sequence of real numbers. Show $\lim_{k \rightarrow \infty} \frac 1 k \sum_{n=1}^k a_n = \inf a_n$

Let $\{a_n\}_{n=1}^{\infty}$ be a monotonic decreasing sequence of positive real numbers. Show that $\lim_{k \rightarrow \infty} \frac 1 k \sum_{n=1}^k a_n = \inf a_n$ Suppose we take the sum of the first $k =m$ numbers then the average will be…
Shuzheng
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