# Tagged Questions

**1**

vote

**0**answers

46 views

### Convergence and summation of power towers or Ackermann functions

I deleted this at StackExchange as it seems advanced or not very relevant for there and I want to ask here.
Let z be a complex number, $Im(z)\neq 0$ or $Re(z)<4$, m be a natural number, let ...

**3**

votes

**2**answers

282 views

### If two functions are equal to their Newton series, is their composition also equal to its Newton series?

Suppose we have two real-valued functions $f(x)$ and $g(x)$, both equal to their Newton series expansion:
$$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)$$
$$g(x) = \sum_{k=0}^\infty ...

**12**

votes

**1**answer

262 views

### Cesaro(?)/Euler(?) - summation of the $s(p)=\sum_{k=0}^\infty (-1)^{H(k)} (1+k)^p$ for $p=1,2,3,…$ (where $H(k)$ is the Hamming-weight)

In another thread (in MO) there was a question about a series where the signs at the terms alternate with the "Hamming-weight", that means according to the number of bits in the binary representation ...

**3**

votes

**3**answers

282 views

### Sum of series $a^{i^2}$

Is there any closed form known for the expression $\sum_{i=1}^\infty a^{i^2}$ where $|a|<1$? Thanks!

**1**

vote

**2**answers

242 views

### What is known about $\displaystyle \sum_k{a^{b^k}}$?

What is known about $\displaystyle \sum_k{a^{b^k}}$? I am very interested in the possible applications of this series.
I have asked about this on Mathematics Stack Exchange here.
I'm wondering if ...

**1**

vote

**2**answers

209 views

### Looking for a limit related to the series in a previous post

Can any one show that the following limit?
$$
\lim_{z\rightarrow \infty} \sqrt{z} \: e^{-z}\sum_{k=1}^\infty \frac{z^k}{k! \sqrt{k}} \quad \stackrel{?}{=} \quad\sqrt{2}-1.
$$
If one uses the ...

**2**

votes

**2**answers

326 views

### How to integrate an exponential function of an exponential function?

Does any one know how to calculate the following integration?
$$
\int_{\mathbb{R}} \left(\exp(z \: e^{-y^2})-1\right)^2 dy=?,\quad z>0.
$$
This post is related to my previous question here , ...

**9**

votes

**1**answer

812 views

### Has anyone seen this series?

I come across the following infinite series.
$$
\sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}.
$$
In particular, I am interested in the case where $a=1/4$.
...

**0**

votes

**1**answer

88 views

### Energy of repeated filter

For given sequences $a=(a_1, a_2, \cdots)$ and $b=(b_1, b_2, \cdots)$, define
$$a \star b$$ as the convolution. Formally, $$c=a \star b$$ implies the $i$th element of $c$, $c_i$, satisfies the ...

**3**

votes

**1**answer

143 views

### Infinite series - analytical solution

Analytical Solution is required for:
$$\sum_{n=0}^\infty (2n+1)\exp(-n(n+1)x),$$
$$\sum_{n=0}^\infty (2n+1)^2\exp(-n(n+1)x),$$
$$\sum_{n=0}^\infty n(n+1)(2n+1)\exp(-n(n+1)x),$$
$$\sum_{n=0}^\infty ...

**7**

votes

**1**answer

410 views

### Is the mapping $f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}$ surjective?

Is the mapping
$$
f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}
$$
surjective?
If not, what is its image?
If yes, what can be said about ...

**1**

vote

**0**answers

75 views

### The $d$-dimension extension of Bernoulli Polynomial

It is known that Bernoulli polynomial has the following Fourier expansion:
\begin{equation*}
B_{2n}(x) = \frac{(-1)^{n-1}2(2n)!}{(2\pi)^{2n}}\sum_{k=1}^{\infty}\frac{\cos(2k\pi x)}{k^{2n}}.
...

**10**

votes

**3**answers

812 views

### Convergent subsequence of $\sin n$

It is well known (not to me -- ed.) that for every real number $\theta \in [0, 1]$ there exists a sequence $(k_i)$ such that $\lim\sin k_i = \theta,$ but there appear to be no explicit such ...

**6**

votes

**1**answer

238 views

### Certain asymptotics involving double infinite sum

Let $1<\alpha<\beta<3/2$. Set
$$
S(n)= \sum_{i,j>0} [i^\alpha+j^\beta]^{-1}[(i+n)^\alpha+(j+n)^\beta]^{-1}.
$$
One can check that $S(n)$ is finite. My question is when $n\rightarrow ...

**10**

votes

**3**answers

549 views

### Calculus Teaching: Is it possible or desirable to give a severely abbreviated treatment of series convergence tests?

I will be teaching Calculus 2 this fall at a large U.S. state university. Our incoming students tend to have a limited or inconsistent background, which limits the amount of material we can cover.
...

**3**

votes

**0**answers

204 views

### The functional equation of Hofstadter's Q sequence

Hofstadter's Q sequence is defined by $Q(1) = Q(2) = 1$ and
$Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$ for $n \geq 3$. So far hardly anything
on this sequence has been proved -- not even that $Q(n)$ is ...

**5**

votes

**3**answers

720 views

### On the continuity of $\sum_{n=1}^{\infty} sin(nx) / n^\alpha$

I know that the series $\sum_{n=1}^\infty \sin(nx) / n^\alpha$, with $0 < \alpha < 1$, converges for $x \in [0,2\pi]$.
I'm trying to understand if the function $f(x) = \sum_{n=1}^\infty ...

**11**

votes

**1**answer

189 views

### An elementary expression for $_3F_2(1,1,9/4;2,2;-1)$

Consider the following series:
$$S=\sum_{n=1}^\infty\frac{(-1)^n\ \Gamma\left(\frac{5}{4}+n\right)}{n^2\ \Gamma(n)}.$$
It can be expressed in terms of a hypergeometric function:
...

**0**

votes

**0**answers

68 views

### Equally subspacing the support of a monotone function, maintaining its mean

SETUP:
Assume $f(\cdot)$ is continuous and strictly monotone decreasing on $[0,\infty]$, with $f(0)>0$ and $f(\infty)<0$.
Let $x_m$ be the solution of $\frac{1}{m}\sum_{i=1}^{m}f(ix)=0$, where ...

**13**

votes

**1**answer

2k views

### The unreasonable effectiveness of Pade approximation

I am trying to get an intuitive feel for why the Pade approximation works so well. Given a truncated Taylor/Maclaurin series it "extrapolates" it beyond the radius of convergence. But what I can't ...

**3**

votes

**2**answers

278 views

### Given a sequence of real numbers,do the following conditions suffice to guarantee convergence to 0?

If $x_{a+1}$-$x_{a}$ converges to $0$ and $x_{2a}$-$2x_{a}$ converges to $0$ , does that imply $x_a$ converges to $0$?

**1**

vote

**1**answer

160 views

### On methods for dealing with recursively defined sequences

Define $a_1=8$ and $a_n=\frac{4^{n+1}-2^{n+2}\sqrt{4^n-a_{n-1}}}{2}$ for $n\geq 2$.
By means of harmonic analysis methods it can be shown that $a_n$ converges to $\pi^2$ (this being the first ...

**1**

vote

**2**answers

479 views

### Convergence of Newton series for sin ax

Let's define half discrete-analytic function as a function whose Newton series converges to that function for each $x>0$:
$$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k f\left ...

**0**

votes

**1**answer

111 views

### An operator realizing the Borel transform

Let $y(z) = \sum_k y_k z^k$ be a holomorphic function in a vicinity of the point $z=0$. Define its Borel transform $By$ as a function $By(z) = \sum_k \frac {y_k}{k!} z^k$.
The well-know formula ...

**6**

votes

**2**answers

410 views

### Alternating sums of GCDs

The exciting question on alternating sums of binomial coefficients triggered me to ask the following much simpler question (sorry if it is too simple, but I'm not a number theorist, so I must be ...

**36**

votes

**2**answers

3k views

### Alternating sum of square roots of binomial coefficients

Let
$$ c_n = \sum_{r=0}^n (-1)^r \sqrt{\binom{n}{r}}. $$
It is clear that $c_n = 0$ if $n$ is odd. Remarkably, it appears that despite the huge positive and negative contributions in the sum ...

**63**

votes

**7**answers

5k views

### Is $ \sum\limits_{n=0}^\infty x^n / \sqrt{n!} $ positive?

Is $$ \sum_{n=0}^\infty {x^n \over \sqrt{n!}} > 0 $$ for all real $x$?
(I think it is.) If so, how would one prove this? (To confirm: This is the power
series for $e^x$, except with the ...

**3**

votes

**1**answer

302 views

### Products of trigonometric functions with increasing frequencies

I am looking at weighted $L_2$ norms of a class of Littlewood polynomials, related to Walsh and Rademacher functions which made me look for pseudo-closed forms or computationally efficient expressions ...

**5**

votes

**2**answers

722 views

### Divergence of Dirichlet series

Suppose $s$ is a complex number with $\Re(s) \in (0,1]$ and $\{a_n\}$ is a complex sequence converging to $a \neq 0$. Must the Dirichlet series $$\sum_{n=1}^\infty\frac{a_n}{n^s}$$ diverge?
I asked ...

**7**

votes

**2**answers

756 views

### Exact Value of a Series

It is very easy to show that the series
$$\frac{1-1/2}{1\times2} - \frac{1-1/2+1/3}{2\times3} + \frac{1-1/2+1/3-1/4}{3\times4} - ...$$
i.e.
$$\sum_{n=1}^{\infty} ...

**2**

votes

**1**answer

314 views

### Can averaged limits of sequences be realized as limits of sequences?

Let a summation method take a sequence $(x_n)$ to a net $(y_\alpha)$, where $\alpha$ runs over a partially ordered set, $y_\alpha=\sum c_{\alpha,n}x_n$ ($c_{\alpha,n}\geq 0$, $\sum_n c_{\alpha,n}=1$ ...

**9**

votes

**3**answers

2k views

### Ramanujan's eccentric Integral formula

The wikipedia page on Srinivasa Ramanujan gives a very strange formula:
Ramanujan: If $0 < a < b + \frac{1}{2}$ then, $$\int\limits_{0}^{\infty} \frac{ 1 + x^{2}/(b+1)^{2}}{ 1 + ...

**11**

votes

**1**answer

2k views

### Ramanujan's Incorrect formula

I actually looked at one of my Questions (posted at MATH.SE) again and found a formula which actually Ramanujan had discovered.
Ramanujan: If $\alpha$ and $\beta$ are positive numbers such that ...

**0**

votes

**2**answers

743 views

### Functions defined as infinite products

Are there standard references on infinite products of rational functions and their convergence properties? I'd appreciate information on finite products too!
The original motivation for this is the ...

**1**

vote

**1**answer

391 views

### Coefficient bounds of an inequality

Hello,
Given positive integers $k$ and $n$. Are there upper bounds on coefficients $A$ and $B$ such that they depends only on $k$ (eg., $2 k^k$) and for all non-negative integer sequences ...

**5**

votes

**2**answers

449 views

### Pseudo-alternate series

Suppose $(a_n)$ is a non-increasing sequence of positive real numbers and $\varepsilon_i = \{\pm 1\},\ \forall i \in \mathbb{N}$ such that $\sum\limits_{i=1}^\infty \varepsilon_i a_i$ is convergent.
...

**0**

votes

**3**answers

1k views

### Inverting a power series? … Cornish Fisher

Hello
In the derivation of the cornish fisher expansion, the following equation is obtained:
$$
\sum_{n=2}^{\infty} b_n H_{n-1}(x_\alpha) = \sum_{j=1}^{\infty}\frac{(x_\alpha - ...

**10**

votes

**3**answers

577 views

### Convergence of alternating harmonic sums

I owe the idea of asking this question to Max Muller and
his curiosity.
What is the set of $\alpha$ in the interval $0\le\alpha < 1$ for which
the alternating sum
$$
...

**45**

votes

**4**answers

2k views

### Nonexistence of boundary between convergent and divergent series?

The following is a FAQ that I sometimes get asked, and it occurred to me that I do not have an answer that I am completely satisfied with. In Rudin's Principles of Mathematical Analysis, following ...

**7**

votes

**1**answer

600 views

### How to rearrange only negative part of a conditionally convergent series to get any sum greater then initial?

Suppose that $\sum^\infty_{n=1} u_n = s,$ where the series converges conditionally, and $s'>s$. How to prove the existence of such a permutation $\sigma,$ such that
1) $u_n\geq 0 \rightarrow ...

**0**

votes

**1**answer

513 views

### Series of squared Fourier coefficients

Hi, if the Fourier series development of $g(t)$ (periodic, $C^\infty$) is
$$
g(t)=\sum_{-\infty}^{+\infty}a_n e^{in\omega t}
$$
does the series
$$
\sum_{-\infty}^{+\infty}\frac{a_n^2}{n^2}?
$$
...

**1**

vote

**0**answers

392 views

### A transformation of infinite series

Suppose I have a convergent infinite series $\sum_{n=0}^\infty (-1)^n a_n = S_0$ and $0 < S_0 < 1$. Write $s_n$ for the $n$-th partial sum. ($s_n = \sum_{k=0}^n (-1)^k a_k$) Now consider the ...

**4**

votes

**2**answers

2k views

### On the behaviour of $\sin(n!\pi x)$ when $x$ is irrational.

Hi,
I'm interested in the behaviour of the sequence $(\sin(n!\pi x))$, when $x$ is irrational, as $n$ tends to infinity.
1) Is the sequence dense in $(-1,1)$?
or
2) Is it possible that for some ...

**19**

votes

**4**answers

1k views

### Can a conditionally convergent series of vectors be rearranged to give any limit?

Warmup (you've probably seen this before)
Suppose $\sum_{n\ge 1} a_n$ is a conditionally convergent series of real numbers, then by rearranging the terms, you can make "the same series" converge to ...

**2**

votes

**1**answer

256 views

### Can we find an l-2 sequence if we know all l-p norms?

I'm wondering if there is a way to approximate the first $M$ terms of a non-increasing $\ell^2$ sequence $\{c_n\}$ if we know
$|c|_p^p = \sum c_n^p$ for $p=2,3,4,\dots$?
I've tried truncating the ...

**6**

votes

**7**answers

633 views

### Are there Generalisations of a Limit (for Just-divergent Sequences)?

There are certain sequences such as
0, 1, 0, 1, 0, 1, 0, 1, ...
that do not converge, but that may be assigned a generalised limit. Such a sequence is said to diverge, although in this case a phrase ...

**8**

votes

**3**answers

1k views

### Which sequences can be extended to analytic functions? (e. g., Ackermann's function)

Let {an} be a sequence of complex numbers indexed by the positive integers. Does there always exist an analytic function f such that f(n) = an for n=1,2,...? If not, are there any simple necessary or ...