Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.

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19
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256 views
+50

Evaluating the sum $\lim_{n\to \infty}\sqrt[2]{2+\sqrt[3]{2+\sqrt[4]{2+\cdots+\sqrt[n]{2}}}}$

The following nested radical $$\lim_{n\to \infty}\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}$$ is known to converge to 2. We can consider a similar nested radical where the degree of the radicals increases: ...
11
votes
0answers
574 views

Are there unique solutions for $n=\sum_{j=1}^{g(k)} a_j^k$?

Edward Waring, asks whether for every natural number $n$ there exists an associated positive integer s such that every natural number is the sum of at most $s$ $k$th powers of natural numbers ...
8
votes
0answers
134 views

Is there a closed form for this sum?

While generalizing the previous result, I conjectured that the series expansion of \begin{align*} \int_{0}^{\frac{\pi}{2}} \arctan \left( \frac{2x \sin\theta}{1-x^{2}} \right) \arctan \left( \frac{2y ...
7
votes
0answers
283 views

How find this sum $\sum_{n=1}^{\infty}\dfrac{L_{n}(2)}{n^4}=?$

Question: Find the sum $$I=\sum_{n=1}^{\infty}\dfrac{L_{n}(2)}{n^4}=?$$ where $$L_{n}(k)=1-\dfrac{1}{2^k}+\dfrac{1}{3^k}-\cdots+\dfrac{(-1)^{n-1}}{n^k}$$ since ...
7
votes
0answers
181 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) ...
6
votes
0answers
63 views

Series with $k^2$ coefficients

Let $\{S_k\}$ be a sequence of numbers. Then by reversing the sum we have the following, $$A=\sum_{k=0}^n k\cdot S_k \cdot S_{n-k} = \sum_{k=0}^n (n-k) \cdot S_{n-k} \cdot S_{k} .$$ Thus adding these ...
5
votes
0answers
110 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
408 views

Proof for an identity involving a sum of binomial coefficients

I am moving through a On The Average Height of Planted Plane Trees by Knuth, de Bruijn and Rice, 1972), and I would like to trade a weaker result for simpler mathematical tools, because my skills are ...
5
votes
0answers
88 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 $M$?
4
votes
0answers
127 views

Proving an equation involving binomial coefficients

Prove that $$\sum_{q=0}^v \binom{v}{q}\frac{q!}{v^{q+1}} = \sum_{q=0}^{v-1} \binom{v-1}{q} \frac{(q+2)!}{v^{q+2}}$$ Thanks. Below are what I have tried: Approach 1: $$\sum_{q=0}^{v-1} ...
4
votes
0answers
137 views

Integral $I=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx$

Hi I am trying to integrate and obtain a closed form result for $$ I:=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx. $$ Here is what I tried (but I do not think this is ...
4
votes
0answers
94 views

Simplifying a sum?

Define polynomials $P_{j,s}^{(r)}$ via the generating series $$\left(\frac{d^s}{dz^s}f(z)\right)^r=\sum_{j=0}^{\infty} P_{j,s}^{(r)}z^j,$$ where $r\geq 1$. Here, $f(z)=z+a_2z^2+a_3z^3+\cdots.$ I was ...
4
votes
0answers
109 views

Difficult Sum: Any Tips?

I'm trying to integrate a very difficult expression, and I've arrived at a particular step involving this sum, which I want to turn into a closed expression so that I can raise it to a power, re-sum ...
4
votes
0answers
105 views

Binomial-like sum involving falling factorials

We know that $\sum_{k=0}^n a^k \frac{n^{\underline k}}{k!} = (1+a)^n$. Is there a known (preferably closed) form for $\sum_{k=0}^n a^k n^{\underline k}$?
4
votes
0answers
77 views

Sums over the positive integers

Can anyone see why it is that if $a$ is large, then $$\log (\sum_m\sum_n \exp(-knm/a)))$$ where $k$ is a constant and $n,m$ take values $1,2,3,...$, can be approximated by $${a\pi^2\over 6k }$$? ...
3
votes
0answers
70 views

Closed form $\sum_{n=2}^{\infty} \frac{1}{\ln^n{n}}$ and $\sum_{n=2}^{\infty} \frac{n}{\ln^n{n}}$

Apologies if this has been asked before, but I was playing around with Wolfram Alpha and got approximations but not closed forms for $$\sum_{n=2}^{\infty} \frac{1}{\ln^n{n}} \approx 3.2426094109 $$ ...
3
votes
0answers
58 views

Evaluation of the double sum

Is there a way to get a closed expression for the double sum: $$\sum\limits_{n = 1}^\infty \sum\limits_{m = 1}^\infty \left( \frac{m}{(n^2 + m^2 - E_1)(n^2 + m^2 - E_2)} \right)^2$$ , where $E_1$ and ...
3
votes
0answers
107 views

Proving $\sum_{k=1}^{n}\binom{n-1}{k-1}{\binom{n+k}{k}}^{-1}=\frac 12$ combinatorially

Question : How can we prove the following equations combinatorially? $$\begin{eqnarray}\sum_{k=1}^{n}\frac{\binom{n-1}{k-1}}{\binom{n+k}{k}}&=&\frac ...
3
votes
0answers
44 views

Finding sum for function with floor-function

I am trying to find a formula to calculate the following sum: $$\sum_{x=0}^n (2ax - {1 \over 2}a^2 - {1 \over 2} a) $$ where $$ a = \left\lfloor {x \over \phi^2} \right\rfloor $$ and $$\phi = {1 + ...
3
votes
0answers
38 views

How to show that $\sum\limits_{k=0}^{\lfloor0.999n\rfloor}\binom{2n}{k} < \binom{2n}{n} $ holds for large n

It seems logical to me since $\binom{2n}{n}$ is in the middle of the row in pascal triangle; therefore, the largest, and for large n the sum adds only the small ones on the left. But I do not have any ...
3
votes
0answers
60 views

Find generating function For sequences

Can anyone out here help? The exercise says: "Find the generating function for each of the sequences below (the general term is given)" Now, the question is how do you find one for those: a) $U_n = ...
3
votes
0answers
74 views

Product of Summations for All Subsets

We have a set $X$ of $n$ integers $\{$$x_1$, $x_2$, .. , $x_n$$\}$, for which there are $2^n$ total subsets. The summation $s$ of a subset $X'$ is simply the sum of all integers present in $X'$, ...
3
votes
0answers
44 views

Showing that a number is part of sequence A000275 in OEIS

Consider the sequence of integers defined recursively by $c_0 = 1$ and $$ c_p = \sum_{l = 0}^{p-1} (-1)^{p+l+1} \binom{p}{l}^2 c_l $$ for $p \geq 1$. This is sequence A000275 in the online ...
3
votes
0answers
161 views

Proving that the finite sum of the each reciprocal of any sequence of integers with common difference is not an integer.

Question : Could you show me how to prove that $\sum_{j=1}^{n}\frac{1}{a+jd}$ is not an integer for any integers $a\gt1, d\gt0$. A week ago, I found the following question in a book: Prove that ...
3
votes
0answers
111 views

Why does the following equation hold?

$\sum_1^\infty\frac{(k\theta e^{-\theta})^k}{k!}=\frac{\theta}{1-\theta}$, where $0<\theta<1$. It can be verified via simulation, but I haven't proved it. Are there any previous results on ...
3
votes
0answers
110 views

Prove the divergence of the sum of the reciprocals of practical numbers

A practical number is an integer $n$ for which every smaller integer can be expressed as a sum of distinct divisors of $n$. How can we prove that the sum of their reciprocals is divergent? Are there ...
3
votes
0answers
90 views

Sum formula for the $\Omega$ constant

I was looking a bit around, and was interested in the konstant $\Omega$. It is defined as number satisfying the equation $$ x e^x = 1 $$ Now, Wikipedia, gives an reccurence relation for the constant ...
3
votes
0answers
59 views

Limit of a sum (no probabilities)

Show that $$\lim_{n\to+\infty}\left(\frac{2}{3}\right)^n\sum_{k=0}^{[n/3]}\binom{n}{k}2^{-k}=\frac{1}{2}$$ without using probabilities. $[\;\cdot\;]$ denotes the integer part.
3
votes
0answers
207 views

Proving identity involving sum

I'm stuck trying to prove the following identity: $$\displaystyle\sum_{i=j}^k\frac{(-1)^{i+j}{k+1 \choose i}{i \choose j}(4S-i-k-3)!(4S-2i-1)}{(4S-i-j-1)!} $$ $$=\frac{(-1)^{j+k}(4S-2k-3)!{k+1 ...
3
votes
0answers
120 views

Bernoulli formula

The sum: $$S_m(n) = 1^m + 2^m + 3^m + 4^m + 5^m...+ n^m$$ Can be calculated by this formula, called the "Bernoulli formula" in wikipedia $$S_m(n) = \frac{1}{m+1}\sum_{k=0}^m {m+1\choose k}B_k ...
3
votes
0answers
271 views

How to calculate the asymptotic expansion of $\sum \sqrt{k}$?

Denote $u_n:=\sum_{k=1}^n \sqrt{k}$. We can easily see that $$ k^{1/2} = \frac{2}{3} (k^{3/2} - (k-1)^{3/2}) + O(k^{-1/2}),$$ hence $\sum_1^n \sqrt{k} = \frac{2}{3}n^{3/2} + O(n^{1/2})$, because ...
3
votes
0answers
82 views

How to sum the series $\sum_{i=1}^n \theta^{a(i)}$, where $a(i)=i(i-1)/2$

This is my very first post, and I was wondering how we can sum the following series $$S_n = \sum_{i=1}^n \theta^{a(i)},$$ where $a(i)=i(i-1)/2$. I don't know if this can help, but $a(i+1) = ...
3
votes
0answers
66 views

Is it possible to algebraically prove that the $n$th degree Taylor remainder of $f(x)$ is less than $K|\Delta x|^{n+1}$ for $K \in \mathbb{R^+}$?

I found a purely algebraic proof, given below, that for a mononomial $f(x) = x^n$ the magnitude of the error of its linear approximation $| f(x) - [f(a) + f'(a)(x-a)] |$ is less than $K(x-a)^2$ for ...
3
votes
0answers
107 views

Limit involving sums of the Von-Mangoldt function

Can someone show that the limit bellow approaches 1/2? Can you also prove that it does, with out using the prime number theorem? $$ \lim_{n\to\infty} \frac{\sum\limits_{k=1}^n \Lambda(k) ...
2
votes
0answers
86 views

Find the limit of $\sum \frac{1}{log^n(n)}$

Working on convergence and divergence of infinite series, I recently focused my attention on the summation $$\displaystyle\sum\limits_{n=2}^{\infty} \frac{1}{log^n(n)}$$ While proving the convergence ...
2
votes
0answers
34 views

statements about summation

Could you help me prove this statements about summation? I know that the second prove is easy of be written, but can I put that summation before cos(theta) and sin(theta)? yes. But why? Do you ...
2
votes
0answers
24 views

Interchanging index of summation in $d$ dimensions

Let $\alpha = (\alpha_{1}, \ldots, \alpha_{d}) \in \mathbb{Z}_{\geq 0}^{d}$ and let $|\alpha| = \alpha_{1} + \cdots + \alpha_{d}$. I have the following question about interchanging summations: Is ...
2
votes
0answers
28 views

Inverse logarithmic integral

If the expansion of the logarithmic interval is$$\text{li}(n) = \log \log n + \gamma + \sum_{k=1}^\infty \dfrac{(\log n)^k}{k! k}$$ what is the inverse of the function?
2
votes
0answers
63 views

Finite Trigonometric Sum

I have a dynamical system model whose equilibria depend on the solution of the following finite sum: \begin{align} \sum_{j\neq ...
2
votes
0answers
36 views

Lower bound of $\sum_{k = 1}^{N}1/(x + k)$

Let $f(x) := \sum_{k = 1}^{N}1/|x + k|$ for $x \in [0, N]$. Why is $f(x) \geq C\log N$ for all $x \in [0, N]$ where $C$ is an absolute constant. My work is: Since $x \in [0, N]$, we can remove the ...
2
votes
0answers
49 views

Interchange of limiting operations (question from an engineer)

I need to clarify when are the below operations valid. If possible, please link me to the related theorems, where I can find details. 1- Given a double integral \begin{equation} \int_{X}\int_Y ...
2
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0answers
33 views

Understanding the summation and floor

I am beginner and a novice for sums and math in general. I don't know the steps or techniques to understand how does the right hand side equals the left? Could someone clarify the techniques or steps ...
2
votes
0answers
42 views

Summation involving Fibonacci numbers

Find: $$ \sum_{n=0}^\infty \sum_{k=0}^n \frac{F_{2k}F_{n-k}}{10^n} $$ where $F_n$ is $n$-th Fibonacci number.
2
votes
0answers
37 views

Sum of Binomials times Logarithms

Is there a closed-form expression or a very good approximation for $$ \sum_{i=0}^n \binom{n}{i} \log (i+1) \,? $$ If the summands alternate, then there is a very close approximation, yet it feels ...
2
votes
0answers
60 views

Find the sum of exponentails of squares $\sum_{r=1}^n e^{-\alpha r^2}$

I would like to find $$a_n =\sum_{r=1}^n e^{-\alpha r^2},\qquad \alpha\in\mathbb{R}$$ I tried to solve the equivalent recursion $$a_n=a_{n-1}+e^{-\alpha n^2}\quad(n>0),\qquad a_0=0.$$ with an ...
2
votes
0answers
86 views

How to prove this combinatorial identity

I am wondering how to prove the following identity: $$\sum_{k=0}^r {r-k \choose m} {s \choose k-t} (-1)^{k-t} = {r-t-s \choose r-t-m}$$ It seems that I can negating the upper index of ${s \choose k-t} ...
2
votes
0answers
42 views

calculating sum of a limit of integral

I am trying to calculate the following expression $$ \sum_{m=0}^{\infty} \frac{1}{m!} \lim_{n \to \infty} \int_{\{(x,y):2x^2+y^2<n^2 \}}\left( 1 - \frac{2x^2+y^2}{n^2}\right)^{n^2} x^{2m}dx~dy ...
2
votes
0answers
67 views

Find Gcd summation fast?

Find the value of the summation: $$ val=\left( \sum_{i=1}^a \sum_{j=1}^b\sum_{k=1}^c....\sum_{x=1}^p GCD(i,j,k,..x) \right)$$ Contraints $2\leq$number of summation terms$\leq 500$, $1\leq ...
2
votes
0answers
224 views

How to calculate this triple summation?

I need to calculate the following summation: $$\sum_{j=1}^m\sum_{i=j}^m\sum_{k=j}^m\frac{{m\choose i}{{m-j}\choose{k-j}}}{k\choose j}r^{k-j+i}$$ I do not know if it is a well-known summation or not. ...
2
votes
0answers
24 views

Is there a closed form for $\sum_{k_1 + \cdots+k_m = n} \frac{\Gamma(k_1 + \alpha)}{k_1!} \ldots \frac{\Gamma(k_m + \alpha)}{k_m!}$?

Let $\alpha$ be a positive real number, and $n,m$ two positive integers. Is there a closed form for this sum? $$\sum_{k_1 + \cdots+k_m = n} \frac{\Gamma(k_1 + \alpha)}{k_1!} \ldots \frac{\Gamma(k_m ...