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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interesting series involving sin and nth roots of unity.

I ran across an interesting series involving the square of sine. It was solved in a clever manner I do not quite get. $$\sum_{k=0}^{n-1}\frac{1}{1+8\sin^{2}(\frac{k\pi}{n})}$$ The solution involved the nth roots of…
Cody
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Find the sum of a series

I'm trying to find the sum of the following series: $$\sum^\infty_{n=1}\frac {(x-3)^{2n}}{2n}$$ I tried to "convert" it to a simple geometrical series, but with no luck. Has someone any idea? Thanks for inspiration! My…
Dávid Molnár
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Sequence and series question. Condition for $x_n$ to be an integer

If $x_1$, $x_2$, $x_3,\ldots$ is a sequence such that $$x_n=\frac{x_{n-2}\space x_{n-1}}{2x_{n-2}-x_{n-1}},$$ where $x_i \in \mathbb R$ and $x_i \ne0 $ for all $i\in \mathbb N$ and $n=3$, $4$, $5,\ldots$ How can I establish necessary and…
user11470
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Nature of the serie $\sum\prod_{k=2}^n (2-e^{\frac{1}{k}})$

I'd like to determine the nature of the following serie : $$\sum_{n\ge 2}\prod_{k=2}^n (2-e^{\frac{1}{k}})$$ Let $u_n = \prod_{k=2}^n (2-e^{\frac{1}{k}})$. So I "have": $$\begin{aligned} \ln(u_n) &= \sum \ln(2-e^{1/k}) \\& \approx \sum \ln(1-1/k +…
Sebastien
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Convergence of Cesàro means for a monotonic sequence

If $(a_n)$ is a monotonic sequence and $$ \lim_{n \to \infty} \frac{a_1 + a_2 + \cdots + a_n}{n} $$ exists and is finite, does $a_n$ converge? If so, does it converge to the same limit? I claimed that this was true in an old answer of mine. I…
Antonio Vargas
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Why aren't mathematical series zero-indexed?

We're learning about sequences in calculus class, and I keep assuming they are zero-indexed because of my experience in programming. Why aren't they zero-indexed? Can they be zero-indexed?
Jared Silver
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Proving divergence of a sequence by proving the sequence is increasing.

We define a sequence recursively by $$a_{n+1}=\frac{1}{4}({a_n}^2+a_n+2)~~~~~~~(a_1=3)$$ By showing $a_n$ is increasing prove that $a_n$ does not converge. Not sure how to do this one. I tried showing that $a_{n+1}-a_n>0$ but couldn't get it to work…
Ryan
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A pyramid of numbers

Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed. Suppose we have a triangle of numbers. Atop the triangle is 1: each number below it is determined by summing half of…
Ali
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Solving recurrence relation varying with parity of n

Given a sequence $u_n$ such that $u_1 = 1$ $u_{2n} = n + u_n$ $u_{2n+1} = n^2 + u_nu_{n+1}$ How to solve for closed-form of $u_n$? I really don't know where to start.
Cranky
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The number of birds after $n$ years

Problem A pair of birds can produce between 3 to 4 birds each year. If we start with a pair of birds and if a bird usually lives for three years, then how many birds will we have after $n$ years? What I have done I assumed that the pair of birds…
user214302
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Proving $1+\frac{4}{2^2}+\frac{1}{3^2}+\frac{1}{5^2}+\frac{4}{6^2}+\frac{1}{7^2}+\frac{1}{9^2}+\frac{4}{10^2}+\frac{1}{11^2}+\cdots=\frac{\pi ^2}{4}$

Proving $$1+\frac{4}{2^2}+\frac{1}{3^2}+\frac{1}{5^2}+\frac{4}{6^2}+\frac{1}{7^2}+\frac{1}{9^2}+\frac{4}{10^2}+\frac{1}{11^2}+\cdots=\frac{\pi ^2}{4}$$ Firstly, I thought to prove it by comparison the terms with the terms of $1/n^2$ , but the…
user187581
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I need the proving of $x\log(x)=(\frac{x-1}{x})+\frac{3}{2!}(\frac{x-1}{x})^2+\frac{11}{3!}(\frac{x-1}{x})^3+...\frac{S_{n}}{n!}(\frac{x-1}{x})^n$

I could find the role of Striling Numbers in the natural logarithm function as follows $$x\log(x)=(\frac{x-1}{x})+\frac{3}{2!}(\frac{x-1}{x})^2+\frac{11}{3!}(\frac{x-1}{x})^3+...\frac{S_{n}}{n!}(\frac{x-1}{x})^n$$ Where $S_n$= absolute Striling…
E.H.E
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Is $\lim na_n \to 0$ sufficient for $\sum a_n$ to converge?

Is $\lim na_n \to 0$ sufficient for $\sum a_n$ to converge? Or additional criteria is required? E.g. $a_n$ needs to be positive? Is naïve comparison with $\frac {1}{n^p}$ series justifies that ? Or is there an obvious counterexample ?
jimjim
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Where does the sum of $\sin(n)$ formula come from?

I have seen Lagrange's formula for the sum of $\sin(n)$ from $1$ to $n$ during one of my classes last week, but I never saw how it came to be. I tried googling it to find a proof but couldn't seem to find any as it kept bringing up his other work…
George Redant
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Proving $\sqrt{2}=\prod_{n=1}^{\infty }\left(1+\frac{0.75}{4n^2-1}\right)$

Proving $$\sqrt{2}=\prod_{n=1}^{\infty }\left(1+\frac{0.75}{4n^2-1}\right)$$ By using the numerical calculation I saw that the convergence of product series is slow, so I need the proving. thanks.
E.H.E
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