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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An Infinite Double Summation $\sum_{k=1}^{\infty} \sum_{n=1}^{\infty} \frac{1}{n^2k^2(n+k)^2}$?

While Solving some integral problem, I encountered the following infinite series: $$\displaystyle \sum_{k=1}^{\infty} \sum_{n=1}^{\infty} \frac{1}{n^2k^2(n+k)^2}$$ I have tried many methods including partial fractions... I seek help! Please provide…
Hasan Kassim
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Riemann rearrangement theorem

The Riemann rearrangement theorem states that if $\sum\limits_{n=0} ^{+ \infty} a_n$ is conditionally convergent and $M \in \mathbb{R}$ then there exists a permutation $ \sigma (n) $ such that $\sum\limits_{n=0}^{+ \infty} a_{\sigma(n)} \ =M$. Could…
Hagrid
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How to compute the series $\sum_{n=0}^\infty q^{n^2}$?

Let $q\in (0,1)$. Is there a way of computing the series $$ \sum_{n=0}^\infty q^{n^2} $$ explicitly? Is there at least a nice accurate estimate? All I could get is the estimate $$\sqrt{\frac{\pi}{4\cdot\mathrm{ln}\frac{1}{q}}}\leq\sum_{n=0}^\infty…
Rasmus
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Evaluating the sum $\sum_{n=1}^{\infty} \frac{e - \left(1 + \frac{1}{n}\right)^n}{\sqrt{n}}$

I want to evaluate $$\sum_{n=1}^{\infty} \frac{e - \left(1 + \frac{1}{n}\right)^n}{\sqrt{n}}$$ But I'm not sure how to approach it. Mathematica suggests that it converges pretty slowly and gives something like 2.57... after around 20,000 terms, but…
AndrewG
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Sum the series $\sum_{n = 1}^{\infty}\{\coth (n\pi x) + x^{2}\coth(n\pi/x)\}/n^{3}$

This sum is from Ramanujan's letters to G. H. Hardy and Ramanujan gives the summation formula as \begin{align} &\frac{1}{1^{3}}\left(\coth \pi x + x^{2}\coth\frac{\pi}{x}\right) + \frac{1}{2^{3}}\left(\coth 2\pi x + x^{2}\coth\frac{2\pi}{x}\right)…
Paramanand Singh
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Do runs of every length occur in this string?

In reference to the strings defined here (constructed by repeatedly appending the last "half" of the current string), consider the particular infinite string $s$ generated by starting with $\text{abc}$: $$\begin{align} \quad…
r.e.s.
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Formula for the $n$th term of $1, 2, 2, 3, 3, 3, 4, 4 ,4, 4, 5, ...$

There exists a formula for the $n$th term of this sequence A002024 from the OEIS ("$n$ appears $n$ times") $$1, 2, 2, 3, 3, 3, 4, 4 ,4, 4, 5...$$ which is $$\left \lfloor \frac {1+\sqrt{1+8n}}{2} \right \rfloor.$$ Is there a better formula for the…
Ali Caglayan
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Is the series $\sum_{i=k}^\infty\frac{\sin\left(\frac{x}{i}\right)}{i}$ bounded?

Comparison with $\dfrac{1}{i^2}$ shows that the series $$\sum_{i=k}^\infty\frac{\sin\left(x/i\right)}{i}$$ is convergent. However, the function of $x$ thus obtained appears to be bounded, with the bound approaching zero in $k$. I have no idea how to…
Thomas Anton
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Given a convergent series of positive rationals, how many subseries can converge to the same number?

Let us have a sequence $a_n$ of positive rational numbers, for which $\sum a_n = R \in \mathbb R$. Now suppose $b_n$ is a subsequence of $a_n$ such that $\sum b_n = r < R$. The question is "Can there be an uncountable number of subsequences of $a_n$…
Benji Altman
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How can I get sequence $4,4,2,4,4,2,4,4,2\ldots$ into equation?

How can I write an equation that expresses the nth term of the sequence: $$4, 4, 2, 4, 4, 2, 4, 4, 2, 4, 4, 2,\ldots$$
ben
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25
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If $a_n$ goes to zero, can we find signs $s_n$ such that $\sum s_n a_n$ converges?

Let $a_n$ be a sequence of complex numbers that converge to zero. Can we always find $s_n \in \{-1,1\}$ such that $\sum_{n=1}^{\infty} s_n a_n$ converges? If the $a_n$ are real numbers, we can find such a sequence $s_n$. If the partial sum of the…
Johan
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What is limit of $\sum \limits_{n=0}^{\infty}\frac{1}{(2n)!} $?

What is the limit of the series $1 \over (2n)!$ for n in $[0, \infty)$ ? $$ \sum_{n = 0}^{\infty}{1 \over (2n)!}$$ I've ground out the sum of the 1st 1000 terms to 1000 digits using Python, (see here ), but how would a mathematician calculate the…
NotSuper
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How can you derive $\sin(x) = \sin(x+2\pi)$ from the Taylor series for $\sin(x)$?

\begin{eqnarray*} \sin(x) & = & x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots\\ \sin(x+2π) & = & x + 2\pi - \frac{(x+2π)^3}{3!} + \frac{(x+2π)^5}{5!} - \ldots \\ \end{eqnarray*} Those two series must be equal, but how can you show that by only…
Paul Goodman
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Convergence of the sequence $\frac{1}{n\sin(n)}$

Does the sequence $$\frac{1}{n\sin(n)}$$ converge to $0$ or not? If not, what's the upper limit?
Dongyu Wu
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Is there any other way to prove this fact? (non-existence of slowest diverging series)

Let $a_n>0$ and $S_n=\sum_{k=1}^{n}a_n$. If $\lim_{n \rightarrow \infty}S_n = +\infty$, then $\sum_{n=1}^{\infty}\frac{a_n}{S_n}=+\infty$. I think this is an important example because it tells us that there exists no series which diverge slowest. So…
Laura
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