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Summation of series involving factorials.

I got this question in a maths contest archive and I am completely clueless over how to start. $$\sum_{m=0}^q(n-m){(p+m)!\over m!}= {(p+q+1)! \over q!}\left(\frac{ n}{ p+1}-\frac {q}{p+2}\right)$$ I ...
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Asymptotic behaviour of sum over the inverse japanese symbol

I am interested in the asymptotic behavior of the sum $$\sum_{m=1}^M\frac{1}{\sqrt{m^2+\omega}}$$ for $1>\omega>0$ in the Limit $M\to\infty$ up to order $\mathcal{O}(M^{-1})$. The first thing I ...
Suppose i need to prove that $\frac{1}{2^2}+\frac{1}{3^2}...+\frac{1}{n^2}<1-\frac{1}{n}$ So in the step of $n+1$, the right side becomes $<1-\frac{1}{n+1}$ or is it: $<1-\frac{1}{n}-\frac{1}... 2answers 84 views Calculate two sums:$\sum_{i=1}^{99}\frac{1}{\sqrt{i+1}+\sqrt{i}}$,$\sum_{i=1}^{9999}\frac{1}{(\sqrt{i}+\sqrt{i+1}) (\sqrt[4]{i}+\sqrt[4]{i+1})}$. Calculate $$\sum_{i=1}^{99}\frac{1}{\sqrt{i+1}+\sqrt{i}}$$ I've figured out that the answer is 9 -there is a pattern that I've figured out. I've created a code and solved it... but how could I do it ... 1answer 31 views Sum algebra solving for coefficient Is the following equation solvable for$k$? $$\sum_{i=1}^{n}\frac{x_ie^{kx_i}}{1+e^{kx_i}} = 0$$ 2answers 20 views Interval for summation function? I know that you almost always set domains for a summation function$\left( \sum \right)$, but can you also set an interval for that domain? Say the domain was 1 to 10, could I set an increment of 0.5 ... 0answers 54 views Simplifying a Double Summation Let$f_n(k)$be defined as $$f_n(k)=\sum_{i=1}^n\sum_{j=1}^i\left(\frac{j}{i}\right)^k$$ Can$f_n(k)$be simplifying down to an expression without summations? By simply graphing$f_n(k)$, it is clear ... 1answer 58 views Sum of$x_1^k+x_2^k+\dots+x_n^k$I was recently wondering if there is some quicker way to compute$x_1^k+x_2^k+\dots+x_n^k$for any natural$k$than just exponentiation and adding one-by-one? Thanks in advance. 1answer 44 views Summation of factorial. $$2(\frac{1}{3!\times7!}+\frac{1}{1\times9!})+\frac{1}{5!\times5!}=\frac{2^a}{b!}$$ find$a,b$by some predictions I see$b=10$but what about numerator. I think we have to$\sum {N\choose r}=2^N$but ... 0answers 41 views sum of subset of complex numbers [duplicate] let there be$\{z_1 ,..., z_n\}$a group of complex numbers. Show that there's a subset$J \subset \{1,...n\}$so that $$\lvert \sum_{k \in J}z_k \rvert \ge \frac{1}{4\sqrt2}\sum_{i=1}^n\lvert z_i \... 1answer 52 views Generalizing a Telescoping Sum \sum_{n=1}^\infty \frac{1}{n+k}-\frac{1}{n} I was trying to generalize an integral I found yesterday on this website and ran into the following interesting sum: S_k=\sum_{n=1}^\infty \frac{1}{n+k}-\frac{1}{n}. I have seen this sum come up a ... 5answers 111 views The solution of equation 4+6+8+10+\cdots +x =270 is 15. [closed] The solution of equation 4+6+8+10+\cdots +x =270 is x=15. How can I prove it? I ve tried the geometric sequence but I cannot figure out the pattern. 2answers 63 views the sum of all four digit multiples of 6 The sum of all four digit multiples of 6 is equal to: A. 8~274~489 B. 8~247~498 C. 8~241~996 Can you help me with this question? I've tried$$S_n= \frac{n(a_1+a_n)}{2}$$with ... 1answer 46 views How to find the General expression of \sum_{k=0}^ {\lfloor n/3\rfloor} {n \choose 3k} [duplicate] Well as the title says I'm having problems trying to derive a general expression for this sum which involves cubic roots of unity$$\sum_{k=0}^ {\lfloor \frac n 3\rfloor} {n \choose 3k}$$Need help ... 2answers 81 views proving \frac{1}{n+3}+\frac{1}{n+4}+…+\frac{1}{2n+4}>\frac{1}{2} how can one prove that: \frac{1}{n+3}+\frac{1}{n+4}+...+\frac{1}{2n+4}>\frac{1}{2} For all natural n, without using induction? thank you. 2answers 27 views \sum _{j=0}^{\infty }\binom{-p-1}{j} \bigl( -\frac {x} {1+x}\bigr) ^{j}=? I did try to use geometric series somehow. I have no idea how to evaluate in terms of p and x. 3answers 84 views Formula for \sum_{i\geq 0} i{n \choose 2i}? So I know that \sum_{i\geq 0}{n \choose 2i}=2^{n-1}=\sum_{i\geq 0}{n \choose 2i-1}. However, I need formulas for \sum_{i\geq 0}i{n \choose 2i} and \sum_{i\geq 0}i{n \choose 2i-1}. Can anyone ... 2answers 92 views Summation of Binomial Coefficient: \sum\binom{n+k}{2k} \binom{2k}k \frac{(-1)^l}{k+1} I am trying to solve this summation problem .$$\sum\limits_{k = 0}^\infty {\left( {\begin{array}{*{20}{l}} {n + k}\\ {2k} \end{array}} \right)} \left( {\begin{array}{*{20}{l}} {2k}\\ k \end{array}} \... 0answers 23 views The Green's function of a second order ODE. The Green's function$G(t,t_0)$or a propagator of an Ordinary Differential Equation (ODE) is a solution to that ODE with the right hand side being replaced by a Dirac delta function$\delta(t-t_0). ... 3answers 854 views Probability sum of 5 before sum of 7 Pair of fair die are rolled (independently I hope) infinitely. Find probability sum of 5 appears before sum of 7. 2 approaches: $$P(\text{sum of 5 appears before sum of 7})$$ $$= P(\text{roll 1 ... 0answers 22 views summation involving a hypergeometric 2F2 function im trying to find the closed form for the following \sum_{n=0}^\infty \frac{c^n}{n!}\frac{(a)_n}{(b)_n}\frac{(\alpha+1/2)_n}{(\alpha+3/2)_n}{_2F_2}(-n,1-b-n;1-a-n,1/2;-\frac{d}{c}) \... 2answers 67 views Theorem 3.16. in Analytic Number Theory by Apostol The below texts are from the book Introduction to Analytic Number Theory by Apostol: I have two questions which I couldn't find solutions for them: 1- According to Thm 3.16., \sum_{n\le x} \... 0answers 37 views Convert Gaussian sum to integral? Consider a sum of the following form:$$S=\sum_{n=0}^\infty e^{A-B(n-C)^2}$$with constant A,B,C>0. Is there any way to convert this sum to an integral and evaluate via Gauss integral? Maybe it ... 2answers 314 views Evaluate \int_{0}^{1} \log\left(\frac{x^2-2x-4}{x^2+2x-4}\right) \frac{\mathrm{d}x}{\sqrt{1-x^2}} Evaluate :$$ \int_{0}^{1} \log\left(\dfrac{x^2-2x-4}{x^2+2x-4}\right) \dfrac{\mathrm{d}x}{\sqrt{1-x^2}} $$Introduction : I have a friend on another math platform who proposed a ... 1answer 57 views A sum of squared binomial coefficients I've been wondering how to work out the compact form of the following.$$\sum^{50}_{k=1}\binom{101}{2k+1}^{2}0answers 9 views Joint distribution of sum and summand Let Z_1 and Z_2 be independent random variables with known distributions F(.;\theta_1) and F(.;\theta_2) of the same convolution closed family. Then Y = Z_1 + Z_2 has distribution F(.;\... 2answers 80 views Find \mathbb P (X_1 + \cdots + X_n = 6n-3) A fair die is tossed n times (for large n). Assume tosses are independent. What is the probability that the sum of the face showing is 6n-3? Is there a way to do this without random variables ... 2answers 26 views Adding powers of 2 to create unique integers Is it true that if an integer k can be represented as 2^a+2^b+...2^n, where a, b ... n are the members of a finite subset of N, there is no other way to represent k as a sum of powers of 2?... 1answer 36 views Sum of cosines with a multiplicative factor in the angle and different interval I have found the following formula for the sum of cosines in both here and here. \begin{align} \sum^n_{l=1} \cos \left(\frac{2 \pi l}{n}\right) = 0 \end{align} I would like to know what the sum ... 1answer 42 views finding the limit of a sequence including sigma symbol I have a sequence :\sum_{k=1}^n e^{\frac{k}{n^2}}\times \ln(k+\frac{1}{k})$$I have to find the limit of this sequence , I tried to surround it ,but this not the correct way. 1answer 68 views Summing power series \sum_{n=1}^\infty \frac{(2n-1)!!}{(2n)!!}x^n Lets have series$$\sum_{n=1}^\infty \frac{(2n-1)!!}{(2n)!!}x^n$$Obviously, its convergence radius is 1. I should sum it, but don't know what's up with the double factorial. There is a hint in the ... 0answers 22 views Does anyone know a summation formula for Wilson's Polynomials? Wilson's polynomials are defined as W_n(x^2; a, b, c, d) := (a+b)_n (a+c)_n (a+d)_n {\space}_4F_3(-n, n+a+b+c+d-1, a+ix, a-ix;a+b, a+c, a+d; 1) Does anyone know a summation formula for Wilson's ... 0answers 14 views Infinite exponential sum doubt Hello! I have a couple of doubts regarding a formula seen here :$$\sum _{k=1}^{\infty } \frac {e^{kz}}{k}= -\log (1-e^{z}) /; Re(z)<0$$What would happen if the real part of z Re(z) were equal ... 2answers 50 views Testing convergence of series \sum_{n=3}^\infty\frac{1}{n (\ln(n))^p(\ln\ln(n))^q} Lets have this problem.$$\sum_{n=3}^\infty\frac{1}{n (\ln(n))^p(\ln\ln(n))^q}$$I have rewritten this to a form$$\sum\frac{1}{np'^{\ln\ln(n)}q'^{\ln\ln\ln(n)}}$$For p,q\in\mathbb{R}. Obviously, ... 1answer 117 views How to justify interchange of sum and integral in fourier series? f is the 4-periodic function f(x) = 1 if x \in [0,2) and f(x) = - 1 if x \in [2,4) The Fourier series of f is$$F(t) = {4 \over \pi} \sum_{n=1}^{\infty} {\sin({\pi \over 2}(2n + 1)t) \... 2answers 48 views How to determine the number of integer solutions to this particular case? Consider the equation $$z_1 + z_2 + z_3 + z_4 + z_5 + z_6 = k$$ For:i = 1, \dotsc,6z_iis a positive natural number and they must satisfy the following: \begin{align} z_1 & \ge 4 \\ z_2 ... 1answer 13 views summation notation of elements in several different (sub)sets If there are two different setsA$and$B$, and$A\cap B= \emptyset$, then sum of all elements in both sets might be written as, $$\sum_{a\in A}a+\sum_{b\in B}b$$ What I want to ask is, can I ... 2answers 49 views Finding$\lim_{L \to \infty} \exp{\frac{T}{L}}\sum_{i=1}^L[ \exp{iA + (i-1)B}]$I am working on a problem and I have come up with a formula that I would like to simply. WLOG, it looks like the following:$\exp{\frac{T}{L}}\sum_{i=1}^L[ \exp{iA + (i-1)B}]$Here,$A,B, T$are ... 1answer 97 views Verify$y=x^{1/2}Z_{1/3}\left(2x^{3/2}\right)$is a solution to$y^{\prime\prime}+9xy=0$This question is a sequel to this previous question. As before, some background information is needed first as follows from my textbook: The standard form of Bessel's differential equation is $$x^... 3answers 181 views Intuitive ways to get formula of binomial-like sum Is there an intuitive way, though I am not sure how to find a conceptual proof either, to establish the following identity:$$\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$$for all natural ... 3answers 138 views Calculate \int_0^{1/10}\sum_{k=0}^9 \frac{1}{\sqrt{1+(x+\frac{k}{10})^2}}dx How can we evaluate the following integral:$$\int_0^{1/10}\sum_{k=0}^9 \frac{1}{\sqrt{1+(x+\frac{k}{10})^2}}dx$$I know basically how to calculate by using the substitution x=\tan{\theta}... 3answers 67 views Infinite Sum of Falling Factorial and Power According to Mathematica,$$\sum_{k=0}^\infty \frac{(G+k)_{G-1}}{2^k}=2(G-1)!(2^{G}-1)$$where$$(G+k)_{G-1}=\frac{(G+k)!}{(G+k-G+1)!}=\frac{(G+k)!}{(k+1)!}$$is the falling factorial. I would ... 4answers 172 views Prove if n \in \mathbb N, then \frac{1}{2!}+\cdots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!} [duplicate] Prove if n \in \mathbb N, then \frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\cdots+\frac{n}{\left(n+1\right)!} = 1-\frac{1}{\left(n+1\right)!} So I proved the base case where n=1 and got \frac{1}{2}... 2answers 76 views does anyone know a nice form of the infinite sum \sum_{n=0, m=0}^{\infty} \frac{a^n b^m}{(n+m)!}? I was wondering if anyone on here knows of a closed form or special function for this infinite sum:$$\sum_{n=0, m=0}^{\infty} \frac{a^n b^m}{(n+m)!}$$Or the sum of any non-trivial subset. 1answer 46 views Series expansion of {x} Hello and sorry for my bad English. I am not mathematician, so sorry if this seems a silly question. I've seen this formula regarding the fractional part of a number in Wikipedia, and I would like to ... 0answers 22 views Check if the sum is equal to the polynomial I have the following polynomial$$(1-\alpha)+3\alpha\beta\gamma+4\alpha\beta\gamma[(1-\beta)+(1-\gamma)]+5\alpha\beta\gamma[(1-\beta)^2+(1-\beta)(1-\gamma)+(1-\gamma)^2]+\cdots$$I believe I can ... 2answers 102 views Show that \sum_{r=1}^n r^4=\frac{3n^2+3n-1}5\sum_{r=1}^n r^2 Following from the question here, I was wondering if it's possible show directly that$$\sum_{r=1}^n r^4=\frac{3n^2+3n-1}5\sum_{r=1}^n r^2$$without expanding the summation in full on either side. 0answers 27 views How to get analytical summation of this series? How to get the analytical summation of this series?$$\sum\limits_{n = 2}^{ + \infty } {{\varepsilon ^{n - 1}}\frac{1}{{{n^3}}}\frac{{{d^2}P_n^2\left( {\cos \theta } \right)}}{{d{\theta ^2}}}} = ?$$... 1answer 32 views Evaluating a series with a constant as a bound I'm trying to find an expression to evaluate a series given a bound that is some unknown constant. For example, the simple summation below can be expressed as such:$\sum \limits_{x=1}^{n}x=\frac{n(...
I am looking out to simplify the following double summation in $\mathbb{F}_q[x_1,x_2]$, where $p$ is a prime and $q=p^k$ for some positive integer $k$ and a positive integer $r$ such that \$0 \leq r \...