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Results tagged with sequences-and-series
Search options answers only
not deleted
user 14830
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.
20
votes
Accepted
Infinite series and sum of two squares
Expanding on my comments: $a(n)$ is congruent mod 5 to the number of
two-square representations $n = r^2 + s^2$
(allowing each of the integers $r,s$ to be positive, negative, or zero).
Thus if $n$ has …
- 79.9k
12
votes
Probability of winning game whereby $T+1$ heads in a row of a coin flip is required to win w...
The probability of not winning is
$$
\prod_{T=1}^\infty \left(1 - \frac1{2^T} \right)
= \frac12 \frac34 \frac78 \frac{15}{16} \cdots
= 0.28878809508660 \ldots ;
$$
that's a well-known constant (equal …
- 79.9k
4
votes
Accepted
2D lattice sum with numerator
The $n=4$ sum is accessible as follows. Expand $(j+k)^4$; by symmetry
the odd terms $4j^3k$ and $4jk^3$ cancel out so we need only sum
$(j^4 + 6j^2k^2 + k^4) / (j^2+k^2)^4$. You say you already know …
- 79.9k
13
votes
In search of an alternative proof of a series expansion for $\log 2$
Write the $n$-th term,
$(-1)^{n-1} \!\left/ \bigl(n {2n \choose n} 2^n\bigr) \right.$,
as the definite integral
$$
\frac14 \int_0^1 \left(-\,\frac{x-x^2}{2} \right)^{n-1} dx
$$
using the formula for t …
- 79.9k
49
votes
Calculation of a series
This is the special case $q=3$ of a formula
$$
\qquad\qquad
\sum_{n=1}^\infty \frac{2^n}{q^{2^{n-1}}+1} = \frac{2}{q-1}
\qquad\qquad(*)
$$
which holds for all $q$ such that the sum converges, i.e. suc …
- 79.9k
8
votes
Asymptotic behavior of a certain trigonometric partial sum
The desired inequality should be true iff
$$
c < c_0 := (r - \sqrt{r^2-1})^2
\quad\ \text{where} \quad\
r = \frac{|a|}{2b}
$$
(NB the hypotheses $b>0$ and $a < -2b$ imply $r>1$, so $0 < c_0 < 1$).
Num …
- 2,712
5
votes
Accepted
Approximation of a square with an irrational arithmetic progression
This holds for all $\alpha \in \bf R$. If $\alpha \in \bf Q$ it's easy,
so we may assume $\alpha$ irrational. Divide by $\alpha$ to get
$$
|\alpha^{-1} k^2 - n| < \alpha^{-1} \epsilon.
$$
So, we wan …
- 79.9k
8
votes
Accepted
Express $\int_0^{\pi/2}\{ \operatorname{gd}^{-1}(x)\}dx$ as series of special functions, wit...
The integral is twice Catalan's constant
$$
G = L(1,\chi_4) = 1 - \frac1{3^2} + \frac1{5^2} - \frac1{7^2} + \frac1{9^2} - + \cdots.
$$
This constant can be computed efficiently to high precision, even …
- 79.9k
18
votes
Accepted
Asymptotics of a Bernoulli-number-like function
[Revised and expanded to give the answer for all $k>1$ and incorporate
further terms of an asymptotic expansion as $n \rightarrow \infty$]
Fix $k>1$, and write $a_1=f(1,k)=1$ and
$$
a_n = f(n,k) =
\f …
- 2,821
6
votes
Accepted
Approximate the following series on the euclidean grid
You're surely right that there cannot be a "closed form" for such a series;
but it can still be approximated to any desired precision.
The defining sum
$$
x = x(a)
= \sum_{i=0}^\infty \sum_{j=0}^\in …
- 79.9k
37
votes
Accepted
The function $\sum_{0}^{\infty} x^n/n^n$
[Edited to outline the end of the argument that $f(-M) \rightarrow 0$
(and to correct a few typos etc. while I'm at it)]
Yes, $F(x) \rightarrow 0$ from below as $x \rightarrow -\infty$.
The convergen …
- 30.1k
23
votes
Accepted
No Tonelli or Fubini
Since there's a "number theory" tag, I suggest the quasimodular form
$E_2(\tau)$, defined for $\tau$ in the upper half-plane as a multiple of
$\sum_{m\in\bf Z} {\sum_{n\in\bf Z}}' (m\tau+n)^{-2}$
whe …
- 79.9k
5
votes
Is there a way to express an power law decay as a series of exponentials?
Yes, even a single exponential, because $a b^x$ is the same as $c e^{kx}$ with $a=c$ and $b=e^k$. But "power law" usually means a multiple of $x^{-r}$, not $b^x$. That can't be written as a sum of e …
- 79.9k
14
votes
Accepted
Vanishing of certain periodic series: A question related to $L(1 , \chi) \neq 0$.
Not necessarily. The first counterexample might be
$q=14$ and $f(n)=1, -1, -1, -1, -1, 1, 0, -1, 1, 1, 1, 1, -1, 0$
for $n=1,2,3,\ldots,14$.
- 79.9k
6
votes
On the continuity of $\sum_{n=1}^{\infty} \sin(nx) / n^\alpha$
For small $x>0$ we can write
$$
f(x) = x^{\alpha-1} \cdot x \sum_{n=1}^\infty \frac{\sin nx}{(nx)^\alpha},
$$
which is $x^{\alpha-1}$ times a Riemann sum for
$$
I_\alpha := \int_0^\infty \sin u \frac{ …
- 79.9k