For questions about recurrence relations, convergence tests, and identifying sequences

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31
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0answers
628 views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where ...
16
votes
0answers
226 views

Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes

Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes. I've been thinking about this question, but ...
16
votes
0answers
421 views

When is an infinite product of natural numbers regularizable?

I only recently heard about the concept of $\zeta$-regularization, which allows the evaluation of things like $$\infty !=\mathop{\hat{\prod}}_{k=1}^\infty k = \sqrt{2\pi}$$ and $$\infty ...
11
votes
0answers
138 views

Are the sums $\sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ transcendental?

This question is inspired by my answer to the question "How to compute $\prod_{n=1}^\infty\left(1+\frac{1}{n!}\right)$?". The sums $f(k) = \sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ (for positive integer ...
11
votes
0answers
135 views

Is there any other way to prove this fact? (non-existence of slowest diverging series)

Let $a_n>0$ and $S_n=\sum_{k=1}^{n}a_n$. If $\lim_{n \rightarrow \infty}S_n = +\infty$, then $\sum_{n=1}^{\infty}\frac{a_n}{S_n}=+\infty$. I think this is a important example because it tell us ...
11
votes
0answers
508 views

Quadratic Recurrence Relation

The following sequence appeared in IMC 2012 (a math competition): $$a_1 = \frac{1}{2}, \qquad a_{n+1} = \frac{n a_n^2}{1+(n+1)a_n}$$ I am trying to find an explicit formula for the sequence. It ...
10
votes
0answers
289 views

I can Euler-sum $\sqrt{-\ln(1)}-\sqrt{-\ln(2)}+\sqrt{-\ln(3)}-\cdots$. But how can I do $\sqrt{-\ln(1)}+\sqrt{-\ln(2)}+\sqrt{-\ln(3))}+\cdots$?

This is also related to an older thread in MSE ("what is the half derivative of zeta at zero?") . One of the possible steps in the problem of that thread was to evaluate the series ...
10
votes
0answers
193 views

To how many decimals is $\sum_ {k=1}^\infty \frac{k}{\sqrt{k!}} = \frac{49850839\,\pi}{29567947}$ correct?

Consider: $$\sum_ {k=1}^\infty \frac{k}{\sqrt{k!}} = \frac{49850839\,\pi}{29567947}$$ This is, as far as I'm able to check with my software, correct to at least 167 decimals. If anyone has the ...
10
votes
0answers
247 views

Two questions about weakly convergent series related to $\sin(n^2)$ and Weyl's inequality

By using partial summation and Weyl's inequality, it is not hard to show that the series $\sum_{n\geq 1}\frac{\sin(n^2)}{n}$ is convergent. Is is true that ...
9
votes
0answers
217 views

Pólya and Szegő, Part I, Ch. 4, 174.

The following is a problem proposed in Pólya and Szegő's book "Problems and Theorems in Analysis" Assume that $0<f(x)<x$ and $$f(x)=x-ax^k+bx^\ell+x^\ell \varepsilon(x),\,\;\;\;\lim_{x\to ...
9
votes
0answers
314 views

Asymptotic related to the infinite product of sine

The amount is somewhat complicated ($x$ is a constant): $$S_n=\sum_{k=1}^n\ln\left(1-\frac{\sin^2\big(x/(2n+1)\big)}{\sin^2\big(k\pi/(2n+1)\big)}\right)\tag{*}$$ I want to enrich my handy powerful ...
8
votes
0answers
221 views

Does the sequence $\sin(n!\pi^2)$ converge or diverge?

Does the sequence $\sin(n!\pi^2)$ converge or diverge?
8
votes
0answers
201 views

Generating function solution to previous question $a_{n}=a_{\lfloor n/2\rfloor}+a_{\lfloor n/3 \rfloor}+a_{\lfloor n/6\rfloor}$

In attempting to answer this question, I reduced it to a seemingly simple generating functions question, but after days of work was unable to construct a proof. Since I do not have experience trying ...
8
votes
0answers
141 views

Function $f(x)$, such that $\sum_{n=0}^{\infty} f(n) x^n = f(x)$

Consider a function $f(x)$. Define Taylor series $\sum_{n=0}^{\infty} f(n) x^n$. Is there such a function, other than constant $0$, that $\sum_{n=0}^{\infty} f(n) x^n = f(x)$? The Taylor series of ...
8
votes
0answers
182 views

Invariant functions on the space of finite sequences of reals

Let $S$ be a space of all finite sequences of real numbers (we don't endow it with metric or topology in general). Before asking the main question, some notation. For each $\mathbf s\in S$ we define ...
7
votes
0answers
105 views

How find this series $\sum_{n=1}^{\infty}\frac{1}{n^2H_{n}}$?

Question: This follow series have simple closed form? $$\sum_{n=1}^{\infty}\dfrac{1}{n^2H_{n}}$$ where $$H_{n}=1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{n}$$ I suddenly thought of this ...
7
votes
0answers
133 views

A Ramanujan-like summation: is it correct? Is it extensible?

I'm still exercising with summation-procedures which I try to make correct Ramanujan-summations. Looking at the series $$ s(1/2,2) = (1/2)+(1/2)^4+(1/2)^9+(1/2)^{16}+... $$ and more general $$ s(b,p) ...
7
votes
0answers
151 views

Find the limit $L=\lim_{n\to \infty} \sqrt{\frac{1}{2}+\sqrt[3]{\frac{1}{3}+\cdots+\sqrt[n]{\frac{1}{n}}}}$

Find the limit following: $$L=\lim_{n\to \infty} \sqrt{\frac{1}{2}+\sqrt[3]{\frac{1}{3}+\cdots+\sqrt[n]{\frac{1}{n}}}}$$ P.S I tried find this limit, but it's made me confused.
7
votes
0answers
89 views

On Bailey and Crandall's sum for $\sum_{n=0}^\infty \frac{1}{5^{5n}}\left(\frac{5}{5n+2}+\frac{1}{5n+3}\right)$

On page 20 of "On the Random Character of Fundamental Constant Expansions", Bailey and Crandall gave the rather unusual sum, $$u_2 = \sum_{n=0}^\infty ...
7
votes
0answers
129 views

How to prove that my sum coincides with a sequence on the Online Encyclopedia of Integer Sequences?

I have a topological space $X$ whose reduced $\bmod 2$ Betti numbers (that is to say, the dimension of the $\bmod 2$ reduced homology) I computed to be $$\small \dim \tilde{H}_t(X; {\mathbb{Z}}_2) = ...
7
votes
0answers
208 views

A sequence similar to the Catalan numbers

The $n$-th Catalan number $c_n$ has the closed form $\frac1{n+1}\binom{2n}{n}$ and follows the recursion $c_n = \sum\limits_{i = 0}^{n-1} c_{n-1-i}c_i$ I am interested in the quantity $e_n$ which ...
6
votes
0answers
75 views

Problem 6 - IMO 1985

For every real number $x_1$ construct the sequence $x_1,x_2,x_3,\ldots$ by setting $x_{n+1}=x_n(x_n+\frac{1}{n})$ for each $n \ge 1$. Prove that there exists exactly one value of $x_1$ for which $0 ...
6
votes
0answers
85 views

Convergence of $\sum_{k=1}^{\infty} \left (\sum_{j=1}^{k}\frac 1 j\right)^{-k}$

Does $\displaystyle\sum_{k=1}^{\infty} \left (\sum_{j=1}^{k}\frac 1 j\right)^{-k}$ converges ? Let's call the inner sum $a_k$ such that $\displaystyle\sum_{k=1}^{\infty} (a_k)^{-k}$, applying ...
6
votes
0answers
139 views

Sum with binomial coefficients and a square root

I encountered this sum from working on an integral: $$\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\sqrt{k}$$ I don't think it can be written as a hypergeometric function, because of this square root. Does ...
6
votes
0answers
282 views

Extension of the Jacobi triple product identity

The Jacobi triple product identity is: $$\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} $$ I would like to extend the idea for ...
6
votes
0answers
374 views

On the weak and strong convergence of an iterative sequence

I have some difficulties in the following problem. I would like to thank for all kind help and construction. Let $H$ be an infinite dimensional real Hilbert space and $F: H\rightarrow H$ be a ...
6
votes
0answers
323 views

Convergence/Divergence of infinite series $ \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{1+\left|{\cos n}\right|}}$

It is well known that $ \displaystyle\sum_{n=1}^{\infty} \frac{1}{n}$ is divergent while $ \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{1+\epsilon}}$ is convergent for a fixed positive value of ...
6
votes
0answers
245 views

expansion for $1-|t|$

Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions $$ f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right) $$ I am wondering if one can expand ...
5
votes
0answers
55 views

Forcing series convergence

The following problem was posed by one of my lecturers: $(z_n)$ a null sequence in $\mathbb{C}$. Does there exist $(\epsilon_n)$ with each $\epsilon_n=\pm 1$ such that: $$\sum_n \epsilon_n ...
5
votes
0answers
81 views

Evaluating $\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k \sinh (k \pi)}$

You can evaluate $\displaystyle \sum_{k=1}^{\infty} \frac{(-1)^{k}}{k^{4n-1} \sinh (k \pi)}$ ($n \ge 1$) by considering the function $ \displaystyle f(z) = \frac{\pi \csc \pi z}{z^{4n-1} \sinh \pi z}$ ...
5
votes
0answers
104 views

How prove this inequality $\frac{1}{n!}\sum\limits_{k=0}^{\infty}\frac{k^n}{k!}\ge e(C\ln{n})^{-n}$

Show that: $$\dfrac{e^n}{(\ln{n})^n}\ge \dfrac{1}{n!}\sum_{k=0}^{\infty}\dfrac{k^n}{k!}\ge e(C\ln{n})^{-n},\ n\ge 2$$ where $C>e$ is constant. My try: ...
5
votes
0answers
104 views

Can one use $e^n$ instead of $2^n$ in Cauchy condensation test?

Cauchy condensation test is useful for testing the convergence of infinite series. The test is stated here as follows: for a positive non-increasing sequence $f(n)$, the sum $\sum_{n=1}^\infty f(n)$ ...
5
votes
0answers
119 views

$\alpha <1$, series $n^{-\alpha} \sum _{k=1}^n \sin (k^2)$ is bounded

Could you help me answer the question, if there exists $\alpha <1$ such that series $n^{-\alpha} \sum _{k=1}^n \sin (k^2)$ is bounded?
5
votes
0answers
85 views

Sum over squared index

Is there any way to compute the finite series $$S_M = \sum_{n=1}^{M} r^{n^2}, $$ for some real $r$, integer $M$?
5
votes
0answers
224 views

Determine sum of exponential

I am struggling to find an answer of the following series $$\sum_{i=1}^n \frac{1}{1+\exp(a_i+b_ix)}$$ Any suggestion?
5
votes
0answers
194 views

Partial summation of a harmonic prime square series (Prime zeta functions)

I am trying to find the following series: $S=\displaystyle\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\dfrac{1}{p_ip_j},A\leq p_1 < p_2 < \dots < p_n \leq B, \lbrace A,B\rbrace \in \mathbb{N}$ ...
5
votes
0answers
100 views

Is there always a telescopic series associated with a rational number?

Here is something I thought up while I was bored and my, erm, fish were busy: Given a rational number $p\in(0,1)$, are there always positive integers $n$, $c_m$, and $w_m$ such that ...
5
votes
0answers
112 views

Determinant expression for the power sum

Let $S_{n,r} := \sum_{k=1}^{n} k^r$ be the power sum. On the homepage by W. Hecht (link) I have found the following determinant expression: $$S_{n,r} = (-1)^{r-1} \frac{n(n+1)}{(r+1)!} \det ...
5
votes
0answers
162 views

$2, 5, 13, 17, 29, 421, 401, 53, 281,…,\rightarrow \infty$? $a_{n+1}=\operatorname{ GPF}(qa_n+p)$

I denote by $\operatorname{ GPF}(n)$ the greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$. Is there a way to prove that the sequence $a_{n+1}=\operatorname{ ...
5
votes
0answers
344 views

Is this a relation between the Riemann zeta function and the Prime zeta function?

I am partly repeating myself here again. Is this a correct relation between the Riemann zeta function and the Prime zeta function? $$ \zeta (s) = \sum\limits_{n=1}^{\infty}\frac{1}{n^{s}}$$ ...
5
votes
0answers
294 views

Is there a reasonably simple, non-recursive formula for this sequence?

I have been curious about the following sequence of rational numbers for some time. I identified the numerator and denominators on the Online Encyclopedia of Integer sequences, but there was not much ...
5
votes
0answers
120 views

Shintani cone zeta function

Is there a procedure/algorithm for calculating sums of the form $$ \sum_{n_1,\ldots,n_r >0} \frac1{L_1(n_1,\ldots,n_r)^{m_1} \ldots L_r(n_1,\ldots,n_r)^{m_r}} $$ where $$ L_i(n_1,\ldots, n_r) ...
4
votes
0answers
58 views

Convergence of $\sum_n |\frac{\cos(3^n)}{n}|$

So a recent post asked about convergence of $\sum_n |\frac{\cos(2^n)}{n}|$, and using double-angle formula for $\cos$ it could be shown that for each pair of consecutive terms, at least one term had ...
4
votes
0answers
75 views

Ugly-nice double series

I'm trying to evaluate the following ugly double sum (presented in raw notation as used in my calculations): $\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{4 m \cos \left(\frac{2 \pi m ...
4
votes
0answers
56 views

Asymptotic expansion of $\zeta(s)$

It is well known that $$ \zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}, \quad \Re[s] > 1, \tag{1}$$ but, if $p \leq N$ denotes the primes less than or ...
4
votes
0answers
114 views

Properties of a certain integer sequence

Consider the following sequence: $$a_1=1$$ $$a_n=\text{Number of subsets of } \{a_1,a_2,...,a_{n-1}\} \text{ that sum to } a_{n-1}$$ The first few elements of that sequence are ...
4
votes
0answers
104 views

Could $4+2+4+2+4+2+\cdots = -1 $?

In physics classes, on this StackExchange and even in blogs the sum $1 + 2 + 3 + 4 + \cdots = - \frac{1}{12} $ has been under the microscope. Why does $1+2+3+\dots = {-1\over 12}$? The ...
4
votes
0answers
53 views

Closest Pair between 2 sets of powers

Assume $a$ and $b$ are natural numbers, $A=\{a,a^2,a^3,\cdots\}$ and $B=\{b,b^2,b^3,\cdots\}$, find $\min\left|a_i-b_j\right|$ where $a_i\in A$ and $b_j\in B$. For example, if $a=3$, $b=10$, then ...
4
votes
0answers
124 views

How to sum the divergent series $1+2+2+3+3+3+\cdots$ and $1+3+2+6+5+4+\cdots$?

In a different context, I encountered the divergent series $1+2+2+3+3+3+\cdots$, and I was wondering about its summation. Putting $f(x)=1+2x+2x^2+3x^3+3x^4+3x^5+\cdots$, we have ...
4
votes
0answers
58 views

Prove that $||\sum_{n=0}^{+\infty}{x_n}||\le\sum_{n=0}^{+\infty}||{x_n}||$ when series $\sum_{n=0}^{+\infty}{x_n}$ are absolutely converge?

I think it should be proved that: Since $$||\sum_{n=0}^{N}{x_n}||\le\sum_{n=0}^{N}||{x_n}||$$ so $$\lim_{N\to+\infty}||\sum_{n=0}^{N}{x_n}||\le\lim_{N\to+\infty}\sum_{n=0}^{N}||{x_n}||$$ so ...