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^...
- 109k
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
$$ \sum_{n=1}^\infty ...
- 109k
34
votes
Accepted
Does this expression always vanish?
Recall that
$$
\sum_iA_i^{n-1}\prod_{j\neq i}\frac1{A_i-A_j}=1,
$$
as follows from the Lagrange interpolation of $x^{n-1}$.
Now apply $\prod_i \frac \partial{\partial A_i}$.
We get
$$
\sum_i\left((...
- 22.2k
33
votes
Accepted
Proving $\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\frac{x_{i}}{x_{j}}\right\}\le \frac{9}{14}n^2$？
OK, here is a fairly simple proof that for any positive integer $n$ and any positive real numbers $x_1,\ldots,x_n$,
$$ \sum_{i,j=1}^{n}\left\{\frac{x_i}{x_j}\right\}\leq \frac{9}{14}n^2\,.$$
That the ...
- 59.8k
30
votes
Accepted
The Mompox Sequence: are all its terms different?
The answer is yes. Suppose $a(m)=a(n)$ for some $m<n$. By the definition of the sequence, we have $a(n)=a(m)\prod_{i=m+1}^ni^{s_i}$, where $s_i\in\{1,-1\}$, so that the above product is equal to $1$...
- 27.4k
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 $k\ge0$
$$c_{3k}=0,$$
$$c_{3k+1}= (-1)^{k+1}\...
- 56.6k
21
votes
Proving $\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\frac{x_{i}}{x_{j}}\right\}\le \frac{9}{14}n^2$？
Too long for a comment.
For $c>0$, the following two claims are equivalent.
Claim 1. An inequality $$
\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\dfrac{x_{i}}{x_{j}}\right\}\leqslant cn^2$$
holds for all ...
- 103k
21
votes
Is this formula expressing $\operatorname{erf} \left(\tfrac 1 {\sqrt 2}\right)$ as an infinite series already known?
Let
$$f(x):=\sqrt{\pi } e^{x^2/4} \operatorname{erf}(x/2).$$
Then
$$f'(x)=\tfrac12\, x f(x)+1.$$
So, by the Leibniz formula, for $m\ge1$,
$$f^{(m+1)}(x)=\tfrac12\,x f^{(m)}(x)+\tfrac m2\,f^{(m-1)}(x)$$...
- 118k
18
votes
Accepted
When can a sum be re-signed to converge to any limit?
Let's rearrange so that $a_1\ge a_2\ge a_3\ge \ldots$. Then this works if and only if $a_n\le \sum_{k>n}a_k$ for all $n\ge 1$.
To see this, let me also rephrase the problem as: Can we obtain all $0\...
17
votes
How many digits of $\sqrt{2}$ are known to date?
My (very limited) understanding of the paper
Polynomial Factorization and Nonrandomness of Bits of Algebraic and Some Transcendental Numbers
is that bit strings of algebraic numbers are not "...
- 26.3k
17
votes
Accepted
How many digits of $\sqrt{2}$ are known to date?
In one sense, all of the base-2 digits (or, I guess, "bits") of $\sqrt{2}$ are known because we have a closed-form formula, according to the OEIS: $$\begin{align} a(n) &= \frac{1}{2} - \...
- 1,625
14
votes
Behavior of $\sum_{n=1}^{\infty} (\{n \xi \} - \frac{1}{2})$ for irrational number $\xi$
This is studied in detail in Sums of Fractional Parts of Integer Multiples of an Irrational
The
sum
$$C(N)=\sum_{n=1}^{N} \bigl( \{ n \xi \} - \tfrac{1}{2}\bigr)$$
satisfies $|C(N)|>c\log N$ for ...
- 178k
14
votes
On the finite sum of reciprocal Fibonacci sequences
Let me sketch a proof that this identity holds for big enough $n$. In fact, we can show that
$$\left(\sum_{k = n}^{2n} \frac{1}{F_{2k}}\right)^{-1} = F_{2n-1} + \frac{1}{\varphi \sqrt{5}} + o(1),$$
...
- 5,061
13
votes
Accepted
On 12 cfracs: for Catalan's $K$, Gieseking's $\kappa$, and $\pi^2$, $\pi^3$, plus three for $\zeta(3)$ using Zagier's "six sporadic sequences"
We have $$C_2(-17,-6,-72)=-(5/8)L(\chi_{-3},2)$$ and
$$C_2(10,3,-9)=(1/2)L(\chi_{-3},2)$$ so both are proportional to what you
call Gieseking's constant but which is simply the value at 2 of the
L ...
- 11.7k
12
votes
Probability of winning game whereby $T+1$ heads in a row of a coin flip is required to win where $T$ is the number of cumulative tails flipped
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 ...
- 77.5k
12
votes
Accepted
If $f(x) = \sum_{n=0}^\infty a_n x^n$, then $\int_{-\infty}^\infty f(x^2) dx = \pi i a_{-\frac{1}{2}}$
Let $\phi(z)$ be analytic function defined on the half-plane
$$H(\delta)=\{z\in \mathbb{C}: \operatorname{Re}z\ge -\delta\}$$
for $0<\delta<1$. Suppose that, for some $A<\pi$, $\phi$ ...
- 753
12
votes
Accepted
Closed form of an infinite series
Q: Does the following infinite series have a closed form?
It does, according to Mathematica:
$$\sum_{n=1}^{\infty} {(-1)^n \frac{\Gamma(\frac{1}{3}+\frac{n}{3})}{\Gamma(1+\frac{n}{3})} \sin\left(\frac{...
- 178k
12
votes
How to show that $\log 2(1/2\log 2\log 4 + 1/3\log 3\log 6 + \dotsb) + 1/2\log 2 - 1/3\log 3 + 1/4\log 4 - \dotsb = 1/\log 2$
Observe that the left-hand side is the sum of two convergent series. Let $N\geq 2$ be an integer tending to infinity. Truncate the first series at the $N$-th term and the second series at the $2N$-th ...
- 99.1k
12
votes
Accepted
A conjectured series expression for the Riemann $\xi$-function and/or Completed L-series. Could this be proven?
Your conjecture is true, and follows trivially from the Poisson summation
formula. For instance, use Theorem 9.4.2 in my book joint with K. Belabas "Numerical Algorithms for number theory", ...
- 11.7k
12
votes
Is this formula expressing $\operatorname{erf} \left(\tfrac 1 {\sqrt 2}\right)$ as an infinite series already known?
WolframAlpha returns this result promptly, so it is probably known or straightforward to show. See here.
- 99.1k
11
votes
Accepted
Does a conditionally convergent sum with random signs converge almost surely?
The answer is no, $\sum \epsilon_n a_n$ need not almost surely converge. For instance, with $a_n = \frac{(-1)^n}{\sqrt{n}}$, the random series $\sum \epsilon_n a_n$ converges with probability zero.
...
- 22.9k
11
votes
Accepted
Suitable closed form for the A079501
Yes!
$i$ is the first part in the composition
$j + 1$ is the number of other parts in the composition
$+1$ accounts for the case that there is only one part
Summing over $i$ and $j$, we want to ...
- 726
11
votes
Accepted
Finding $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$
If you plot $\log f_n$ versus $n$, with $f_n=\sum_i x_i^{2n}$, then the asymptotic slope for large $n$ will give you the largest of the $x_i$; subtracting that contribution from $f_n$ and repeating ...
- 178k
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}$$
for $n,q \geq 0$. Indeed, note ...
- 109k
10
votes
Accepted
Asymptotic for Ramanujan's $\tau$-function
While an asymptotic for $|\tau(n)|$ does not exist, there are many results that help us to nail down the order of $|\tau(n)|$. First, let us write
$$\tau(n)=n^{\frac{11}{2}}f(n).$$
Also, let $d(n)$ ...
- 5,059
10
votes
Accepted
How can I evaluate the following sum?
The corresponding infinite sum is related to the "incomplete theta function",
$$\Theta_0(x,y)=\sum_{n\geq 0}x^ny^{n(n-1)/2}.$$
We have
$$\sum_{n\geq 0}a^{n^2}=\Theta_0(a,a^2).$$
Accordingly, ...
- 88.9k
10
votes
Accepted
Binomial series
$\newcommand\ep\varepsilon\newcommand\de\delta$Note that
$$\sum_{k=0}^n k^a\binom nk=n^a2^n E\Big(\frac{X_n}n\Big)^a, \tag{1}\label{1}$$
where $X_n$ is a binomial random variable with parameters $n,1/...
- 118k
9
votes
Accepted
On the finite sum of reciprocal Fibonacci sequences
We need to prove, equivalently
$$\frac1{F_{2n-1}+1}<\sum_{k=n}^{2n}\frac1{F_{2k}}\le \frac1{F_{2n-1}}, $$
that is, by the above expression for $F_{k}$, since $\beta=-\alpha^{-1}$, we need to check ...
- 56.6k
9
votes
Accepted
Let $(a_n)_{n\in N}=(1,2,3,4,6,8,9,12,\cdots)$ list the set$\{2^n3^m\mid m,n\in N\}$. Find $α$ such that $(a_n)\alpha\pmod1$ is not equidistributed
$\alpha =\sum_{n=1}^{\infty} \frac{1}{ 6^{100^n}}$ should do the trick. A positive proportion of numbers on your list are of the form $2^a 3^b$ for $a,b$ within a reasonable constant factor of each ...
- 137k
9
votes
Asymptotic for Ramanujan's $\tau$-function
No, $\tau(n)$ fluctuates wildly, and it cannot be described in simpler terms. It is "irreducible arithmetic data", and we just love that. Same for its absolute value.
- 99.1k
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