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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Contrasting 1×2×3... with 1+2+3...

I was wondering what comparing the graph of 1+2+3... with 1×2×3... would look like. In doing so, I tried graphing 1×2×3... in Excel, but no matter how many points I plotted (e.g. 5, 120 vs. 8, 40320) I kept getting a jagged angle towards the end,…
Sam
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Optimizing the decay rate of a particular sequence

There is a sequence of numbers $a_0, a_1, a_2, \ldots$ which I would like to approach zero as fast as possible. This sequence evolves according to the recursion $$ a_{t+1} = (1-0.001 b_t)a_t + 50 b_t^2,$$ where $b_t$ is a sequence I have control…
robinson
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Determine the convergence of a series.

Here is the series: $$ \sum_{n = 1}^{\infty}\frac{\sqrt{n + \sqrt{n + \sqrt{n}}}}{(n + (n + n^2)^2)^2}$$ The method I use to determine this series is comparison test which is that I construct the following sequence : $$ a_n =…
oscarmetal break
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Given $\alpha>0$. Prove that eventually $e^{n^{\alpha}}>n$, for $n \in \mathbb{N}$

Given $\alpha>0$. Prove that there exists $N \in \mathbb{N}$ such that $e^{n^{\alpha}}>n$, for $n \in \mathbb{N}$, $n \geq N$.
math123
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Row sum and column sum of double series $a_{m,n}=\frac{m-n}{2^{m+n}}\frac{(m+n-1)!}{m!n!}$

Show that given a double series $a_{m,n}=\frac{m-n}{2^{m+n}}\frac{(m+n-1)!}{m!n!}$, where $a_{0,0}$ is defined to be zero, its sum by rows $\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}a_{m,n}$ is -1, its sum by columns…
Charlie Chang
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Range of Convergence of $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n-1}}{n \ 3^n (x-5)^n}$

$$\sum\limits_{n=1}^{\infty} \frac{(-1)^{n-1}}{n \ 3^n (x-5)^n}$$ I am trying to use the alternating series test to find a range of $x$ for which $(1) b_n > b_{n+1}$ and $ (2) \lim_{n \to \infty} \frac{1}{n \ 3^n (x-5)^n} = 0$. If $|\frac{1}{x-5}|…
A. Radek Martinez
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Infinite series question requiring no explanation

Determine if the statement is true or false. No explanation needed. $$\sum_{n=0}^{\infty}\frac{\sin n}{n!}\leq e$$ Although no explanation is needed I was wondering how you would approach this problem in the first place. Could I possibly use a…
EhBabay
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Are these infinite sum identities on the real unit interval new?

Let $a,b \in [0,1]$. Set $a_1 = a$, $b_1 = b$, and define inductively $a_{n+1} = a_n(1-b_n)$ and $b_{n+1} = b_n(1-a_n)$. Then $$\sum_n^\infty a_nb_n = \text{min}(a,b)$$ We have proven this result for $\omega$-complete effect monoids (of which…
John
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Solution to an infinite series

Given an infinite series in the form of: $$a \cdot a^{2\log(x)} \cdot a^{4\log^2(x)} \cdot a^{8\log^3(x)} \dotsb = \frac{1}{a^7} $$ find the solution for all positive and real $a$ other than $1$. The textbook, from which this problem was taken,…
SierraTango
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What is the sequence (1, 7, 2, 22, 3, 11, 4, ...)?

What is this sequence? I was told, that every mathematician would know this sequence, because it's subject of research. Does anyone recognize it? Thanks in advance, Florian
Florian
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What kind of series is this, and how do I sum it?

$\displaystyle\sum{\frac1{a_n}}$, e.g. $$\frac 1 1 + \frac 1 2 + \frac 1 3 + \frac 1 4 + \dots + \frac 1 n$$ or $$\frac 1 2 + \frac 1 4 + \frac 1 6 + \frac 1 8 + \dots + \frac 1 {2n}$$
Shawn
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Show that a conditionally convergent sequence has divergent sign subsequences.

In my real analysis class, we learned how to show that an absolutely convergent series, namely $\sum\limits_{n=1}^{\infty}{a_n}$, has convergent subsequences $\sum\limits_{n=1}^{\infty}{a_n^+}$ and $\sum\limits_{n=1}^{\infty}{a_n^-}$. Now, I'm being…
Neurax
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$u_n=\frac {1}{1\cdot n}+\frac {1}{2(n-1)}+\ldots+\frac {1} {n \cdot 1}$,then $\lim u_n=?$

I am stuck on the following problem and do not know to tackle it: If $u_n=\dfrac {1}{1\cdot n}+\dfrac {1}{2 \cdot (n-1)}+\dfrac {1}{3 \cdot (n-2)} + \ldots +\dfrac {1}{n\cdot1}$, then $\lim u_n = 0$. Can someone point me in the right direction? …
user53386
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Evaluating $\lim\limits_{n\to\infty}\frac1{n^4}·\left[1·\sum\limits_{k=1}^nk+2·\sum\limits_{k=1}^{n-1}k+3·\sum\limits_{k=1}^{n-2}k+\cdots+n·1\right]$

Find the value of$$L=\lim_{n\to\infty}\frac1{n^4}\cdot\left[1\cdot\sum_{k=1}^nk+2\cdot\sum_{k=1}^{n-1}k+3\cdot\sum_{k=1}^{n-2}k+\cdots+n\cdot1\right].$$ My attempt is as…
user3290550
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Order property of infinite series

I had a question about a result that I found myself trying to use often. Suppose $\{a_n\}$ and $\{b_n\}$ are sequences in $[0,\infty]$ such that $a_n \leq b_n$ for all $n \in \mathbb{N}$. Then, \begin{equation} \sum_{n = 1}^\infty a_n \leq \sum_{n…
Colin Jackson
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