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.
65,956
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What are some mathematically interesting computations involving matrices?
I am helping designing a course module that teaches basic python programming to applied math undergraduates. As a result, I'm looking for examples of mathematically interesting computations involving ...
- 4,408
66
votes
12
answers
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Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2} + \cdots $
I know that the harmonic series $$\sum_{k=1}^{\infty}\frac{1}{k} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \cdots + \frac{1}{n} + \cdots \tag{I}$$ diverges,...
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Is $ \sum\limits_{n=1}^\infty \frac{|\sin n|^n}n$ convergent？
Is the series
$$ \sum_{n=1}^\infty \frac{|\sin n|^n}n\tag{1}$$
convergent？
If one want to use Abel's test, is
$$ \sum_{n=1}^\infty |\sin n|^n\tag{2}$$
convergent？
Thank you very much
- 1,073
66
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2
answers
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When is an infinite product of natural numbers regularizable?
I only recently heard about the concept of $\zeta$-regularization, which allows the evaluation of things like
$$\infty != \prod_{k=1}^\infty k = \sqrt{2\pi}$$
and
$$\infty \# = \prod_{k=1}^\infty ...
- 1,029
66
votes
1
answer
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Are the sums $\sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ transcendental?
This question is inspired
by my answer to the question
"How to compute $\prod_{n=1}^\infty\left(1+\frac{1}{n!}\right)$?".
The sums
$f(k) = \sum_{n=1}^{\infty} \frac{1}{(n!)^k}$
(for positive integer $...
- 108k
65
votes
7
answers
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How can I show that $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt\ldots}}}$ exists?
I would like to investigate the convergence of
$$\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt\ldots}}}}$$
Or more precisely, let $$\begin{align}
a_1 & = \sqrt 1\\
a_2 & = \sqrt{1+\sqrt2}\\
a_3 &...
- 65.8k
65
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2
answers
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Does $\sum _{n=1}^{\infty } \frac{\sin(\text{ln}(n))}{n}$ converge?
Does $\sum _{n=1}^{\infty } \dfrac{\sin(\text{ln}(n))}{n}$ converge?
My hypothesis is that it doesn't , but I don't know how to prove it. $ζ(1+i)$ does not converge but it doesn't solve problem here.
- 1,354
65
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2
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What is the average rational number?
Let $Q=\mathbb Q \cap(0,1)= \{r_1,r_2,\ldots\}$ be the rational numbers in $(0,1)$ listed out so we can count them. Define $x_n=\frac{1}{n}\sum_{k=1}^nr_n$ to be the average of the first $n$ rational ...
- 6,360
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3
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Review of my T-shirt design [closed]
I'm a graphics guy and a wanna-be mathematician. Is the T-shirt design below okay? Or if there's a bone headed error, I'd appreciate a heads up.
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- 833
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answer
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Is OEIS A248049 an integer sequence?
The OEIS sequence A248049 is defined by
$$ a_n \!=\! \frac{(a_{n-1}\!+\!a_{n-2})(a_{n-2}\!+\!a_{n-3})}{a_{n-4}} \;\text{with }\; a_0\!=\!2, a_1\!=\!a_2\!=\!a_3\!=\!1.$$
is apparently an integer ...
- 36k
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votes
12
answers
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$\lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite
How do I prove that $ \displaystyle\lim_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite?
- 1,648
61
votes
3
answers
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Proving $\left(1-\frac13+\frac15-\frac17+\cdots\right)^2=\frac38\left(\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\cdots\right)$
The equality$$\left(1-\frac13+\frac15-\frac17+\cdots\right)^2=\frac38\left(\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\cdots\right)\tag{1}$$follows from the fact that the sum of the first series ...
- 431k
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answer
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Is there a "good" reason why $\left\lfloor \frac{n!}{11e}\right\rfloor$ is always even?
(A follow-up of sorts to this question.)
The quantity $\left\lfloor \frac{n!}{11e}\right\rfloor$ is always even, which can be proved as follows.
Using the sum for $\frac{1}{e}$, we split the ...
- 38.2k
60
votes
8
answers
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Is there a common symbol for concatenating two (finite) sequences?
Say we have two finite sequences $X = (x_0,...,x_n)$ and $Y = (y_0,...,y_n)$.
Is there a more or less common notation for the concatenation of these sequences, like $\sum (X,Y) = (x_0,...,x_n,y_0,...,...
- 1,431
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votes
4
answers
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Proving $\frac{\sin x}{x} =\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{2^2\pi^2}\right) \left(1-\frac{x^2}{3^2\pi^2}\right)\cdots$
How to prove the following product?
$$\frac{\sin(x)}{x}=
\left(1+\frac{x}{\pi}\right)
\left(1-\frac{x}{\pi}\right)
\left(1+\frac{x}{2\pi}\right)
\left(1-\frac{x}{2\pi}\right)
\left(1+\frac{x}{3\pi}\...
- 2,201