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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Find the $n^{th}$ term of this sequence

Given the following sequence of numbers 1, 3, 7, 9, 13, 15, 21, 25, 27, 31, 33, 37, 43, 45, 49, 51, 55, 57, 63, 67, 69, 73, 75, 79, 85, 91, 93, 97, 99 Can anyone show me the calulation for the nth term? The sequence of first differences can be…
Ron
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How prove exists a sequence $\{a_{n}\}$ of real numbers such that $\sum_{n=1}^{\infty}a^2_{n}<\infty,\sum_{n=1}^{\infty}|a_{n}b_{n}|=\infty$

Suppose that the series $\displaystyle\sum_{n=1}^{\infty}b^2_{n}$ of postive numbers diverges. Prove that there exists a sequence $\{a_{n}\}$ of real numbers such…
math110
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A sequence such that the sum of any $p$ successive terms is negative, and the sum of any $q$ successive terms is positive

Question : Let $p,q \in\mathbb N$ which satisfy $p\lt q$. In a finite sequence of real numbers, let us consider a sequence such that the sum of any $p$ successive terms is negative, and the sum of any $q$ successive terms is positive. Then, can…
mathlove
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Formula for $1^2+2^2+3^2+...+n^2$

In example to get formula for $1^2+2^2+3^2+...+n^2$ they express $f(n)$ as: $$f(n)=an^3+bn^2+cn+d$$ also known that $f(0)=0$, $f(1)=1$, $f(2)=5$ and $f(3)=14$ Then this values are inserted into function, we get system of equations solve them and get…
Templar
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Find coefficient of $x^{50}$ in $(\sum_{i=1}^{\infty}x^n)^3$

How do you find the coefficient of $x^{50}$ in $(\sum_{i=1}^{\infty}x^n)^3$?
James Lee
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General term formula of $S_n=\sum\limits_{i=1}^n{\frac{a^i}{i!}}$?

If $S_n=\sum\limits_{i=1}^n{\frac{a^i}{i!}}$, where a is a positive number, then what's the general term formula of $S_n$?
Jim
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How find the $a_{n}$ close form

Question: let $a_{1}=1,a_{2}=3$, and $a_{n+2}=(n+1)a_{n+1}+a_{n}$ find the close form $a_{n}$ my try: let $$\dfrac{a_{n+2}}{(n+1)!}=\dfrac{a_{n+1}}{n!}+\dfrac{a_{n}}{(n+1)!}$$ so …
user94270
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Elementary proof that $\sum_n 1/(p_n \log p_n)$ converges for primes $p_n$

The prime number theorem says that the $n$th prime number is $p_n = \Theta(n \log n)$, so the series $\sum_n 1/(p_n \log p_n)$ should converge by comparison to $\sum_n 1/n (\log n)^2$. However this seems like overkill using deep mathematics. Is…
user2566092
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Find all positive integers $n$ for which $1 + 5a_n.a_{n + 1}$ is a perfect square.

The sequence $a_1, a_2, \ldots $ is defined by the initial conditions $$a_1 = 20; \quad a_2 = 30$$ and the recursion $$a_{n+2} = 3a_{n+1} - a_n$$ and for $n \geq 1$. Find all positive integers $n$ for which $1 + 5a_n * a_{n+1}$ is a perfect…
Arshdeep Singh
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Explosive number(s) from $x_{n+1}=\left(1+\frac{1}{x_n}\right)^n$

I came accross a curiosity I don't fully understand. In an exam, the sequence $\left(x_n\right)$ is introduced such as for all $n \geq 1$ $$ x_{n+1}=\left(1+\frac{1}{x_n}\right)^n $$ with $x_1 = \alpha \in \left]0;+\infty\right[$. It is argued that…
Atmos
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What does 1/(2 + 3/(4 + 5/(6 + 7...))) converge to?

I'd like to find a closed-form expression for $$ x = \frac{1}{\displaystyle2 + \frac{3}{\displaystyle4 + \frac5{6+\cdots}}}$$ My own attempt implies $ x = \frac{1}{2} $. But by numerical approximation the sequence seems to converge to $ \approx…
gudetama cutey
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Find the number of terms common in two sequences

How many terms do the two sequences $S_1$ and $S_2$ have in common? $S_1 = 1, 3, 6, 10, 15\dots$ up to $200$ terms. $S_2 = 3, 6, 9, 12, 15\dots$ up to $200$ terms. I need to know the number of common terms in these two sequences (irrespective of…
Walter White
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How to evaluate $\sum_{n=1}^\infty \frac{n^5}{e^{2\pi n}-1}$?

I recently found the following result on Twitter. $$\sum_{n=1}^\infty \frac{n^5}{e^{2\pi n}-1}=\frac{1}{504}$$ I know that $\int_0^\infty \frac{x^5}{e^{2 \pi x}-1} dx = \frac{5!}{(2\pi)^6}\zeta(6)=\frac{1}{504}$ How to show that the sum is also…
Archisman Panigrahi
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Closed-form of $ S_n = 2^n p^2 + 2^{n-1} p^4 + 2^{n-2} p^8\cdots + 2p^{2^n}, $ where $n$ is positive integer?

Does, there exist some closed-form solution of the following finite-series ? $ S_n = 2^n p^2 + 2^{n-1} p^4 + 2^{n-2} p^8\cdots + 2p^{2^n}, $ where $n$ is a Positive Integer and $0
kaka
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$(n+1)x_{n+2}-nx_{n+1}-x_n=0$ ,prove the sequence converges

Sequence $x_n$ for which $$(n+1)x_{n+2}= nx_{n+1}+x_n$$ for every $n\in\mathbb{N}$. Prove that it converges. Its not decreasing or increasing, i checked with some random initial values.So, i dont know how to proceed with this. Any help?
Plom
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