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Results tagged with sequences-and-series
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user 766
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.
9
votes
Must bounded sequences be well-distributed to most *composite* moduli?
This question is related to the results in
Bergelson, Vitaly; Richter, Florian K., Dynamical generalizations of the prime number theorem and disjointness of additive and multiplicative semigroup actio …
- 114k
10
votes
Accepted
Remarkable recursions for the A204262
I can show the first identity $R(n,0) = f_{n+1,n+1}(0)$, as a consequence of the more general identity
$$ R(n,q) = \frac{1}{(q+1)!} f_{n+q+1,n}(n+1)\tag{1}\label{1}$$
for $n,q \geq 0$. Indeed, note t …
- 114k
37
votes
Accepted
Elegant recursion for A301897
Here is an expanded version of the generating function argument I sketched in a comment.
For $i=1,2,3$, define the generating functions $F_i(x,y) := \sum_{n=0}^\infty \sum_{q=0}^\infty R(n,3q+i) x^n y …
- 114k
34
votes
Accepted
Can we just use the linear term of exponential sums to sum divergent series
This summation method gives answers that are close to, but do not always match, traditional divergent summation methods. For instance, for constants $a,b>0$, the divergent sum
$$ \sum_{n=1}^\infty (a …
- 114k
101
votes
A number theory problem where pi appears surprisingly
Note that $a_k$ is always a multiple of $k$ (in particular, this structure disrupts the random model proposed in comments). We can exploit this structure to simplify the recurrence and clarify the dy …
- 114k
11
votes
Accepted
Is this infinite product entire?
This function is (on the real line, at least) the product of
$$ \exp( \mu^2 \sum_{i=1}^\infty |z_i|^2 - 2 \mu \Re(\sum_{i=1}^\infty z_i)) \quad (1)$$
and the Hadamard type product
$$ \prod_{i=1}^\inft …
- 1,439
11
votes
Discrete entropy of the integer part of a random variable
Using (say) decimal notation, ASCII encoding, and a delimiter symbol such as a space or comma, as well as the law of large numbers, one can almost surely encode $N$ independent copies of $\lfloor X \r …
- 114k
16
votes
What are the properties of this polynomial sequence?
Here is an algebraic formalism that seems to finish off the problem. Let ${\mathbb Z}[x]$ be the ring of integer polynomials in $x$, and let ${\mathbb Z}[x]^{\mathbb N}$ denote the ring of functions …
- 114k
209
votes
Accepted
Is the series $\sum_n|\sin n|^n/n$ convergent?
Note that if $\pi$ were rational (with even numerator), then $\sin(n)$ would equal $1$ periodically, so the series would diverge. Similarly if $\pi$ were a sufficiently strong Liouville number. Thus …
- 114k
5
votes
Accepted
$\sum\limits_{n=1}^{\infty}(-1)^n ((\frac{a_{n+1}}{a_{n}})^2-1)$ converges.Does $\sum\limits...
No, one can create sequences in which the first sequence converges but the second does not (or vice versa).
For sake of argument take $k=2$. To begin with let us ignore the requirement that the $a_n …
- 114k
65
votes
Accepted
Is $\sum_{k=1}^{n} \sin(k^2)$ bounded by a constant $M$?
No. If one selects a number $k$ at random from $1$ to a large number $n$, then for any fixed $h$, the random variables $\sin((k+1)^2), \dots, \sin((k+h)^2)$ asymptotically have mean zero, variance 1/ …
- 114k