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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Find limit of this recursive sequence

$$ a_0=0,\ a_1=2,\ a_{n+1}=\sqrt{2 - \frac{a_{n-1}}{a_n}} \\ \lim_{n\to\infty}2^na_n\ =\ ? $$
morning glory
  • 149
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Does $\sum_{k=0}^\infty \frac{16(2k+1)^2}{(4m^2 - (2k+1)^2)^2}=\pi^2$?

I think the following series has the value $$\sum_{k=0}^\infty \frac{16(2k+1)^2}{(4m^2 - (2k+1)^2)^2}=\pi^2$$ for any positive integer $m$, but I am not sure how to go about computing it. Plugging the above sum into a computer for a few different…
Jay
  • 329
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Does $\sum\frac{\sin n}{n}$ converge?

Does $\sum\frac{\sin n}{n}$ converge? I have tried the comparison test, root test and ratio test but still can't prove it is convergent or divergent.
john
  • 283
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Limit of $y=\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-\sqrt{5+\ldots}}}}}$

I would appreciate any help with this problem: If $$y=\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-\sqrt{5+\ldots}}}}}$$ Then how do I find $y^2 - y$? I'm not sure whether this is an arithmetic or geometric series.
Quaxton Hale
  • 1,258
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Is $a_n=\sum_{\alpha=0}^n\;\text{round}(\sin\alpha)$ bounded?

Is the sequence $(a_n)$ bounded if $$a_n=\sum_{\alpha=0}^n\;\text{round}(\sin\alpha)?$$ Edit: round$(x)$ simply means round $x$ to the nearest integer; the case where $x=\pm0.5$ may be ignored since $\pi$ is irrational. Please see comments for our…
Supware
  • 940
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Prove that $a_{n}=0$ for all $n$, if $\sum a_{kn}=0$ for all $k\geq 1$

Let $\sum a_{n}$ be an absolutely convergent series such that $$\sum a_{kn}=0$$ for all $k\geq 1$. Help me prove that $a_{n}=0$ for all $n$. Thank you!
student
  • 141
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Is there a closed form for the series $\sum_{k=1}^\infty \frac{\ln(4k-3)}{(4k-3)}-\frac{\ln(4k-1)}{(4k-1)}?$

This is actually a question which was deleted by a previous user. I worked very hard on this question and became quite engrossed. The question was simply: Is there a closed form for the series $$\sum_{k=1}^\infty…
000
  • 5,760
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Show that:$\sum\limits_{n=1}^{\infty}{n\over (4n^2-1)(16n^2-1)}={1\over 12}(1-\ln{2})$

Show that $$\sum_{n=1}^{\infty}{n\over (4n^2-1)(16n^2-1)}={1\over 12}(1-\ln{2})$$ My try: We split into partial decomposition $$n={A\over 2n-1}+{B\over 2n+1}+{C\over 4n-1}+{D\over 4n+1}$$ Setting $n={1\over 2}$, ${-1\over2}$ we have $A={1\over3}$…
gymbvghjkgkjkhgfkl
  • 8,090
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Calculation of a strange series

Is it possible to find an expression for: $$S(N)=\sum_{k=0}^{+\infty}\frac{1}{\sum_{n=0}^{N}k^n}?$$ For $N=1$ we have $$S(1) = \displaystyle\sum_{k=0}^{+\infty}\frac{1}{1 + k} = \displaystyle\sum_{k=1}^{+\infty}\frac{1}{k}$$ which is the (divergent)…
Riccardo.Alestra
  • 10,546
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Closed form for $\sum\limits_{n=1}^{\infty}\frac{n^{4k-1}}{e^{n\pi}-1}-16^k\sum\limits_{n=1}^{\infty}\frac{n^{4k-1}}{e^{4n\pi}-1}$

We took the idea from this Ramanujan's identity $$\frac{1^{13}}{e^{2\pi}-1}+\frac{2^{13}}{e^{4\pi}-1}+\frac{3^{13}}{e^{6\pi}-1}+\cdots=\frac{1}{24}$$ A few examples of Ramanujan-type…
user339807
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2 answers

an infinite series involving odd zeta

I ran across a cool series I have been trying to chip away at. $$\sum_{k=1}^{\infty}\frac{\zeta(2k+1)-1}{k+2}=\frac{-\gamma}{2}-6\ln(A)+\ln(2)+\frac{7}{6}\approx 0.0786\ldots$$ where A = the Glaisher-Kinkelin constant. Numerically, it is approx.…
Cody
  • 14,054
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Evaluating $\sum_{n=1}^{\infty} \frac{1}{n^2+1}$

While I know that $$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^{2}}{6}$$ But trying to evaluate this has left me stumped $$\sum_{n=1}^{\infty} \frac{1}{n^2+1}$$ I evaluated it through wolfram alpha, it gave me $\frac{1}{2}(\pi\coth(\pi)-1)$. What…
iron2man
  • 589
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4 answers

Can $\sum_{k\in M}\frac{1}{k}$ be a large integer?

I'm interested in the following question: Given an integer $n_0$. Is there always an integer $n>n_0$ and a finite subset $M\subset \mathbb N$ with $\sum_{k\in M}\frac{1}{k} = n$. This is not a homework problem, I don't know if there is an easy…
Lukas Betz
  • 4,506
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If $\sum{a_n}$ converges does that imply that $\sum{\frac{a_n}{n}}$ converges?

I know if $\sum{a_n}$ converges absolutely then $\sum{\frac{a_n}{n}}$ converges since $0\le \frac{|a_n|}{n} \le |a_n| $ for all $n$ so it converges absolutely by the basic comparison test and therefore converges. However, I cannot prove the…
ItsNotMeItsYou
  • 836
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Equivalence of Dirichlet density and asymptotic density

Let $A, B$ be subsets of the natural numbers with $A \subseteq B$. The asymptotic density of $A$ in $B$ is $$\lim_{N\to \infty}\frac{\text{number of elements of }A\text{ below }N}{\text{number of elements of }B\text{ below }N}.$$ The Dirichlet…
Anna B
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