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

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4
votes
3answers
47 views

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 ...
10
votes
3answers
281 views

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 ...
2
votes
3answers
48 views

induction clarification about the step $n+1$

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}...
2
votes
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 ...
0
votes
1answer
31 views

Sum algebra solving for coefficient

Is the following equation solvable for $k$? $$\sum_{i=1}^{n}\frac{x_ie^{kx_i}}{1+e^{kx_i}} = 0$$
0
votes
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 ...
2
votes
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 ...
3
votes
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.
1
vote
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 ...
2
votes
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 \...
1
vote
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 ...
1
vote
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.
0
votes
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 $...
0
votes
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 ...
-3
votes
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.
0
votes
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 $x$.
2
votes
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 ...
2
votes
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}} \...
0
votes
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 $\delta(t-t_0)$. ...
6
votes
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 ...
1
vote
0answers
22 views

summation involving a hypergeometric 2F2 function

im trying to find the closed form for the following \begin{equation} \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}) \...
0
votes
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} \...
0
votes
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 ...
11
votes
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 ...
1
vote
1answer
57 views

A sum of squared binomial coefficients

I've been wondering how to work out the compact form of the following. $$\sum^{50}_{k=1}\binom{101}{2k+1}^{2}$$
1
vote
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(.;\...
2
votes
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 ...
-1
votes
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?...
2
votes
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 ...
1
vote
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.
1
vote
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 ...
0
votes
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 ...
0
votes
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 ...
2
votes
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, $...
1
vote
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) \...
-1
votes
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: $i = 1, \dotsc,6$ $z_i$ is a positive natural number and they must satisfy the following: \begin{align} z_1 & \ge 4 \\ z_2 ...
0
votes
1answer
13 views

summation notation of elements in several different (sub)sets

If there are two different sets $A$ 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 ...
0
votes
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 ...
1
vote
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^...
12
votes
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 ...
5
votes
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}...
3
votes
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 ...
3
votes
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}...
2
votes
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.
0
votes
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 ...
0
votes
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 ...
3
votes
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.
0
votes
0answers
27 views

How to get analytical summation of this series?

How to get the analytical summation of this series? $$\sum\limits_{n = 2}^{ + \infty } {{\varepsilon ^{n - 1}}\frac{1}{{{n^3}}}\frac{{{d^2}P_n^2\left( {\cos \theta } \right)}}{{d{\theta ^2}}}} = ?$$ ...
0
votes
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(...
1
vote
1answer
63 views

Simplification of a double summation in a polynomial ring over a finite field

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 \...