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.

65378 questions
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Simpler derivation of $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$

I know that the equality $$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$$ can be proved in numerous ways by using the Fourier series. However, is there a way to derive it using more fundamental tools? I've tried: $$ \sum_{n=1}^\infty…
Adam Latosiński
  • 10,558
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Please suggest a sequence with the following properties

Please suggest a most simple sequence with the following properties: $$\sum_{n=1}^{\infty} a_n=1$$ $$\frac1{a_n} \sim n!$$
Anixx
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Recursive sequences

I have $$s_1 = 1, s_n = ns_{n-1}$$ I don't know what this means at all, sequence 1 equals 1, sequence number = number times sequence subscript number - 1 Is that it? Because it doesn't work at all when I try to work it out.
Adam
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Closed form of a sequence?

I am so terrible at finding a closed form of a given sequence. Please help me on the following sequence: $$ f(x) = 1 -…
Yakup
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Possible average square value

suppose the sum of seven Positive number is 21. what is the minimum possible value of the average of the square of these number?
Gajanand Jamnadas Baraiya
  • 41
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Infinite sum of harmonic number:$\frac12+\left(1+\frac12\right)\frac1{2^2}+\left(1+\frac12+\frac13\right)\frac1{2^3}+\cdots$

I learned that I can find the value of some infinite sum. Then what is the value of this sum? $$\frac12 + \left(1+\frac12\right)\frac1{2^2}+\left(1+\frac12 +\frac13\right)\frac1{2^3}+\left(1+\frac12 +\frac13 +\frac14\right)\frac1{2^4} + \cdots…
S. Yoo
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$a_n > 0$ and $\sum_\limits{n=1}^{+\infty} \frac{1}{a_n}$ converges. Prove $\sum_\limits{n=1}^{+\infty} \frac{n}{a_1 + \cdots + a_n}$ is convergent.

$a_n > 0$ and $\sum_\limits{n=1}^{+\infty} \frac{1}{a_n}$ converges. Prove $\sum_\limits{n=1}^{+\infty} \frac{n}{a_1 + \cdots + a_n}$ is convergent. I find that this may have something to do with Stolz Theorem which says that if $\{\frac{1}{a_n}\}$…
XT Chen
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Sum of Infinite series $\frac{1.3}{2}+\frac{3.5}{2^2}+\frac{5.7}{2^3}+\frac{7.9}{2^4}+......$

Prove that the sum of the infinite series $\frac{1.3}{2}+\frac{3.5}{2^2}+\frac{5.7}{2^3}+\frac{7.9}{2^4}+......$ is 23. My approach I got the following term $S_n=\sum_1^\infty\frac{4n^2}{2^n}-\sum_1^\infty\frac{1}{2^n}$. For…
Samar Imam Zaidi
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How to derive this sequence: $1^3+5^3+3^3=153,16^3+50^3+33^3=165033,166^3+500^3+333^3=166500333,\cdots$?

I found it is interesting but I don't know how the R.H.S is coming from the L.H.S, i.e, how to derive this sequence? The sequence is as…
Suresh
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What values of $a$ make the series $\displaystyle \sum_{n=0}^{\infty}\frac{n!2^n}{e^{an}n^n}$ convergent?

I would like to find what values of $a$ make the following series convergent $$\sum_{n=0}^{\infty}\frac{n!2^n}{e^{\large an}n^n}$$ I started applying $n$-root but i don't know how to solve $$\lim _{n\to +\infty}\frac{\sqrt[n]{n!}}{n}$$ If there are…
user63534
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Maybe there is an interpretation of something like $ S_{-3}$?

I'm doing some exploratory afternoon reading, and I'm baffled by a minor detail in this paper. Here is the background: A sequence $S = S_n$ is almost convergent to L if for any $\epsilon > 0$ we can find an integer n such that the average of n or…
futurebird
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A series transformation

Why the equality $$ \sum_{k=0}^{2n} \sum_{i=\lfloor{\frac{k}{n+1}}\rfloor(k-n)}^{k-\lfloor{\frac{k+1}{n+1}}\rfloor(k-n)} \binom{n}{i} \binom{n}{k-i} $$ $$ = \sum_{k=0}^{2n} \sum_{i=0}^{k} \binom{n}{i} \binom{n}{k-i} $$ holds? I couldn't find any…
user429582
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Product of a and b should be equal to the sum of all numbers in the sequence excluding a and b

I came across the below question in codewars. Regarding to the question, all I know is the sum of numbers ranging from 1 to n is $n(n+1)/2$. And I have no idea how to solve this question further. There were plenty of code solutions available but I…
James K J
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Sum of series $\sum_{k=1}^\infty{-\frac{1}{2^k-1}}$

What is the sum of the series $\displaystyle\sum_{k=1}^\infty{\frac{-1}{2^k-1}}$? Also, more generally, can we find $\displaystyle\sum_{k=1}^\infty{\frac{-1}{c^k-1}}$ for some $c$?
Matt Groff
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Asking about $\sum_{n=2}^{\infty}\frac{(-1)^n}{n}\left[\frac{4372}{(2n-1)^7(n-1)}+\frac{n^2+n+4372}{(2n+1)^7(n+1)}\right]$

$$\sum_{n=2}^{\infty}\frac{(-1)^n}{n}\left[\frac{4372}{(2n-1)^7(n-1)}+\frac{n^2+n+4372}{(2n+1)^7(n+1)}\right]=\frac{61}{184320}\pi^7\tag1$$ Step…
Dragon
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