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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Finding the maximum of $a_1$ given $2S_n=a^2_{n+1}-a_{n+1}$

Let sequence $\{a_n\}$ be$$2S_n=a^2_{n+1}-a_{n+1},\ \forall n\in N^+$$ where $S_n=a_1+a_2+\cdots+a_n$, and $a_2=a_9$, then the maximum of the $a_1$ is: A. $3\qquad\qquad$ B. $6\qquad\qquad$ C. $9\qquad\qquad$ D. $12$ I try…
math110
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Existence of limit in a recurrence equation: $\alpha_{n+2}=\frac{\alpha_{n}+(1/\alpha_{n+1})}{2}$

Let be $\boldsymbol{\alpha_{n+2}=\frac{\alpha_{n}+(1/\alpha_{n+1})}{2}}$ a recurrence equation with known $\alpha_0$ and $\alpha_1$. How do you prove that $\lim_{n\to\infty}\alpha_n$ exists? Note that no conditions are to be assumed about $\alpha_0$…
ksoriano
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Can a bounded number sequence be strictly ascending?

The title says it. Can a bounded number sequence be strictly ascending / descending? I have a problem that tells me the sequence of fractional parts $(\{nx\})_{n\geq 1}$ (where $x$ is given) is ascending. But I know that the sequence is bounded…
furfur
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Limit $a_{n+1}=\frac{(a_n)^2}{6}(n+5)\int_{0}^{3/n}{e^{-2x^2}} \mathrm{d}x$

I need to find the limit when $n$ goes to $\infty$ of $$a_{n+1}=\frac{(a_n)^2}{6}(n+5)\int_{0}^{3/n}{e^{-2x^2}} \mathrm{d}x, \quad a_{1}=\frac{1}{4}$$ Thanks in advance!
user63534
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Finding the general formula of $a_{n+1}=\frac{a_n^2+4}{a_{n-1}}$ with $a_1=1$ and $a_2=5$

Find the general formula of $a_{n+1}=\dfrac{a_n^2+4}{a_{n-1}}$ with $a_1=1$, $a_2=5$. I have tried to write the recursion as a product, make summations, tried to look at patterns but its value grows very fast: $1,5,29,169,985,5741$… So I ran out…
Sebastian Aranvergara
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How to show that $\sum_{i=1}^\infty\frac{1}{i\cdot2^i}=\ln2$?

I have no idea about showing that: $$\sum_{i=1}^\infty\frac{1}{i\cdot2^i}=\ln2$$ And what about more general situation(replace 2 with a constant $\alpha$)? Could anyone please give me a helping hand? Any help would be appreciated.
ArchanJin
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Find $a_n$ via $a_n - a_{n-1}=3^n$

I remember the middle school taught how to find the closed-form expression of $a_n$ given $$ a_0=1 $$ $$ a_n - a_{n-1}=3^n $$ What's name of such sequence? What's general approach to obtain $a_n$? ps. the sequence is $1,4,13,40, \dots$,…
whitegreen
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If $S=1+\frac{1}{2^4}+\frac{1}{3^4}+\frac{1}{4^4}\cdots$, then what is $\lfloor S \rfloor$?

If $$S=1+\frac{1}{2^4}+\frac{1}{3^4}+\frac{1}{4^4}\cdots$$ then $$\lfloor S \rfloor = \text{?}$$ What I tried: I know that $$S=1+\frac{1}{2^4}+\frac{1}{3^4}+\cdots=\zeta(4)=\frac{\pi^4}{90}\approx 1.1$$ then $\lfloor S \rfloor =1$. But…
jacky
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How to prove that $\csc^2x=\sum_{-\infty}^{\infty}\frac{1}{(x-n\pi)^2}$

I was reading the book The Princeton Companion to Mathematics On page $293$, there is a statement $$\csc^2x=\sum_{n=-\infty}^{\infty}\frac{1}{(x-n\pi)^2}\tag{1}$$ Here is one method from the book: Observe first that $$h(x)=…
Larry
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Show that a sequence is bounded

Let $\delta>0$,$f(x)=\frac{1}{5}x+x^{1+\delta}$.Let $x_0>0$,define $x_n=f(x_{n-1})$ for all $n \in \mathbb{N}$.How to show that $\{5^nx_n\}_{n=1}^{\infty}$ is a bounded sequence for all sufficiently small $x_0$?It seems quite obvious that this is…
Ben
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Find the smallest possible value of the sum $x_1+x_2+...+x_{2008}$

Let $x_1, x_2,...,x_{2008}$ are numbers such that $|x_1|=999$ and for all $n=2,...,2008$ $$|x_n|=|x_{n-1}+1|$$ Find the smallest possible value of the sum $$x_1+x_2+...+x_{2008}$$ My work: Let $S=x_1+x_2+...+x_{2008}$. If $x_1=-999, x_2=-998, ...,…
Roman83
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Given $a_n > 0$ and $\sum a_n$ diverges, is this a valid proof that $\sum \frac{a_n}{1+a_n}$ diverges?

I'm aware that similar questions have been posted before, for example here. But my question is not a duplicate since I am asking about my specific proof. Is it correct, and could it have been made simpler while retaining the same basic idea? If…
Brennan Vincent
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Turtle Race (Or: A Sequence of dependent Integer Series)

while trying to implement a parallel QR decomposition, I was faced with a type of "sequences of integer series" which exceeds my mathematical understanding. So I'm asking You for help. For a lack of a better name, I've dubbed the problem the "Turte…
DirkT
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Generalize $\sum_{n=1}^{\infty}\frac{H_n}{(n+1+x)(n+2+x)\cdots(n+k+1+x)}$

I was looking at this wolfram site on section [25] A general identity due to B. Cloitre (pers. comm., Jan. 7, 2006) where $k\ge1$ $$\sum_{n=1}^{\infty}\frac{H_n}{(n+1)(n+2)\cdots(n+k+1)}=\frac{1}{k!k^2}\tag1$$ I was trying to generalize $(1)$ Let…
Endgame
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The sum of the series $\frac{15}{16}+\frac{15}{16} \times \frac{21}{24}+\frac{15}{16} \times \frac{21}{24} \times \frac{27}{32}+\dots$

Suppose $S=\frac{15}{16}+\frac{15}{16} \times \frac{21}{24}+\frac{15}{16} \times \frac{21}{24} \times \frac{27}{32}+\dots \dots$ Does it converge? If so find the sum. What I attempted:- On inspection of the successive terms, it easy to deduce…
user440191
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