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Results tagged with sequences-and-series
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user 6101
for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.
1
vote
Accepted
Series of quotients with perturbed denominator
Since $\sum_ {n=1}^\infty \frac{a_n}{b_n } < \infty$ and $0 \le \frac{a_n}{b_n + \sigma/N}\le \frac{a_n}{b_n} $, we have that $\sum_{n=1}^N \frac{a_n}{b_n + \sigma/N} \to \sum_ {n=1}^\infty \fra …
- 60.6k
3
votes
Two Equal Series?
Since for sequences $\|a\|_p\to \|a\| _\infty$, maximum term is determined by the values of $\sum_i a_i^s$. Arguing inductively on the powers sum of the remaining terms one concludes. (this assuming …
- 60.6k
2
votes
The sum of an hydrogen atom related infinite series
Incidentally, note that the series telescopes, as
$$a_n={\Gamma(n)^2\over\Gamma(n+1/2)\Gamma(n+3/2)}=-4\big[(n-1/2)(n+1/2)-n^2\big]{\Gamma(n)^2\over\Gamma(n+1/2)\Gamma(n+3/2)}=$$
$$=-{4\,\Gamma(n)^2\o …
- 60.6k
8
votes
Non-arithmetic proof of the integrality of a rational expression
The generating function $g(x):=(1−k^2 x)^{-1/k}$ satisfies, besides $g(0)=1,$
$ g^k= 1 + k^2 x\ g^k $,
whence we may express $c(k,n)$ as a sum of products of $c(k,j)$, with $j < n$, showing induc …
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15
votes
Accepted
Convergent subsequence of $\sin n$
As to the convergenge to zero: note that the convergents of the continuous fraction for $\pi$ provide a rational approximation $|\pi - p_n/q_n| < 1/q_nq_{n+1}$ so that $\sin p_n\to 0$.
The sequence o …
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3
votes
Accepted
Calculating "factorial sequence" of a rational function
A comment on the issue of determining the coefficients of the expansion in function series in Fedor Petrov's answer. Recall the generating function of the Stirling numbers of the second kind $S(n,r)$ …
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5
votes
About $\lim_{n\to +\infty} n\prod_{k=1}^{n-1}\cos^2(k)$
It is graphically evident and easy to prove that if for $\theta\in\mathbb R$ one has $\cos^2\theta \ge c:=\cos^2\frac12=0,7701\dotso<1$, then $|\cos^2(\theta+1)|\le c$. As a consequence the number of …
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10
votes
Sums of arctangents
I have no references for this particular series, but here's some hints to get a closed formula for the coefficients listed above by Michael Renardy.
If we let $u:=\frac {1-x} 2$, an expansion
$$\arct …
- 60.6k
7
votes
Radio-playing sequence
Although we have by now a precise answer, I'd like to keep the summer mood of the question and play a little more with it by an elementary arithmetic approach.
The solution I wish to sell is good for …
- 60.6k
4
votes
Accepted
A restricted version of Riemann series theorem: rearrangements with alternating signs
Here you prescribe in addition the sequence of signs of the rearranged series in the Riemann-Dini theorem to be alternating, but note that any non-stationary binary sequence of signs does as well. Mo …
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10
votes
Accepted
How to show this series converges $\sum\limits_{n=1}^\infty n^{-1/2}\sin(n)\sin(n^2)$
As indicated by Todd Trimble in comments, we can use the Dirichlet test; here, since $$\sin(n)\sin(n^2)=\frac12\big( \cos n(n-1) - \cos n(n+1) \big)$$
we have a telescopic sum $$\sum_{n=1}^M \sin(n) …
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22
votes
Closed form of an infinite series
Denote $c_n:={(-1)^n \frac{\Gamma(\frac{1}{3}+\frac{n}{3})}{\Gamma(1+\frac{n}{3})} \sin(\frac{2\pi n}{3})}$ the $n$-th term of the series. We have for all $k\ge0$
$$c_{3k}=0,$$
$$c_{3k+1}= (-1)^{k+1}\ …
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13
votes
Where in mathematics do these polynomials appear?
Not a truly satisfying answer, but maybe it puts things under a slightly more natural view. Consider the linear map $L$ on the space $k[x]$ such that $Lp(x):=p(x^2)$ . So $(L-I)^k$ expands by the bin …
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16
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The function $\sum_{0}^{\infty} x^n/n^n$
I wish to add a remark to Noam D. Elkies' beautiful answer. From the integral representation for $f$, putting $e^{-t}=s$ in the integral,
$$f(-x)=1-x\int_0^\infty e^{-xte^{-t}} e^{-t}dt = 1-x\int_0^1 …
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2
votes
Limit of functions and asymptotic behaviour
Note that the quantity $\lambda p$ depending on the parameter $x$, as you define it for polynomials $p(t):=\sum_{k=0}^m a_k t^k$, that is
$$(\lambda p )(x)=\sum_{k=0}^m a_k \frac{1}{1+k/x}\, , $$
can …
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