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

learn more… | top users | synonyms

0
votes
1answer
41 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
75 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 ...
0
votes
0answers
22 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
849 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
19 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
66 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
36 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 ...
9
votes
2answers
263 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
34 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
41 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
65 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
21 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
113 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 ...
-3
votes
0answers
30 views

Exponential Sum Limit

Im having a little bit of difficulty finding the general solution to the following: $\sum_{i=1}^L \exp({\frac{AB}{L}})^i$ where $A>0$ and $B$ is real. I want to know what happens as $L \...
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
0answers
75 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^...
9
votes
1answer
120 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
134 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
65 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
137 views

Prove if $n \in \mathbb N$, then $\frac{1}{2!}+\cdots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$

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
21 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 ...
2
votes
2answers
99 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
26 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
31 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
62 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 \...
1
vote
2answers
35 views

I need help reindexing the sum

I know this is probably exceedingly simple, but I'm just stuck and keep making some mistake. Here, $t_n$ represents the n-th Tribonacci number. That is, $t_0 = 0, t_1 = 0, t_2 = 1$ and $t_n = t_{n-1}...
2
votes
5answers
102 views

Sum of combinatorics sequence $\binom{n}{1} + \binom{n}{3} +\cdots+ \binom{n}{n-1}$

I need to find sum like $$\binom{n}{1} + \binom{n}{3} +\cdots+ \binom{n}{n-1},\qquad \text{ for even } n$$ Example: Find the sum of $$\binom{20}{1} + \binom{20}{3} +\cdots+ \binom{20}{19}=\ ?$$
4
votes
3answers
213 views

proof of $1^4+2^4+…+n^4=\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}$ [duplicate]

I want a 'simple' proof to show that: $$1^4+2^4+...+n^4=\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}$$ I tried to prove it like the others but I can't and now I really need the proof. Also I want a geometric ...
5
votes
2answers
88 views

Determine $\frac{f''(\frac{1}{2})}{f'(\frac{1}{2})}$ if $f(x) = \sum_{k=0}^{1000} \ {2015 \choose k}\ x^k(1-x)^{2015-k}$

Problem : Determine $\frac{f''(\frac{1}{2})}{f'(\frac{1}{2})}$ if $f(x) = \sum_{k=0}^{1000} \ {2015 \choose k}\ x^k(1-x)^{2015-k}$ Trying to simply brute force the problem, yields the following ...
1
vote
3answers
38 views

Finite sub-sums of finite, countably infinite sums.

Suppose that $I$ is a countable set and $$ \sum_{i \in I} X_i = X,$$ where $X \in \mathbb{R}$ (in particular $X$ is finite). Does this mean that for all $\epsilon >0$ there exists a finite subset ...
1
vote
2answers
54 views

Finding convergence zone/range for $\sum_{i=1}^\infty \frac{x^{n^2}}{n(n+1)}$

$$\sum_{i=1}^\infty \frac{x^{n^2}}{n(n+1)}$$ I used the ratio test and I end up with: $$|x|*\frac{n}{n+2}$$ What steps do I need to take to continue? Looking for hints or steps, not full solution/
3
votes
3answers
143 views

Does this $\lim_{n \to +\infty} \frac{1}{n^2} \sum_{k=1}^{n} k \ln\left( \frac{k^2+n^2}{n^2}\right )$ exist?

I need to examine whether the following limit exists, or not. $$\lim_{n \to +\infty} \frac{1}{n^2} \sum_{k=1}^{n} k \ln\left( \frac{k^2+n^2}{n^2}\right )$$ If it does, I need to calculate its value. ...
0
votes
2answers
45 views

Recurrence Relation with two parameters and Summation

This is a recurrence relation with two parameters which came up in a problem I was trying to solve. Given $$\begin{align}&A_n=pB_{n-1};\qquad &&B_n=q(A_{n-1}+B_{n-1})\\ &A_4=p; \...
0
votes
0answers
35 views

Formula for continuous interest with compounding principal

I'm trying to figure out a formula for compounding interest along with a compounding principal that is added to every month and paid in full (please bear with me as my terminology may be incorrect). I'...
1
vote
3answers
33 views

Proof that $Qo(n) = 2(\sum_{i=1}^{n-1}i)+2n = n^2 + n$

So i would appreciate if someone explain to me the step by step on how do i get this result $Qo(n) = 2(\sum_{i=1}^{n-1}i)+2n = n^2 + n$ How do you proof that it is $=n^2+n$ ?
0
votes
1answer
27 views

Summation Closed form for floor$\left(\log_n\right)$

The closed sum for the floors of logs of consecutive integers is: $$\sum_{i=0}^n \lfloor \log_2i\rfloor = n\lfloor \log_2n\rfloor-2^{\lfloor \log_2n\rfloor+1}+\lfloor \log_2n\rfloor+2$$ This formula ...
4
votes
3answers
77 views

Prove the congruence $ \sum_{r=1}^{p-1}{(r|p) * r } \equiv 0 \pmod p.$

Prove that if $p$ is prime and $p\equiv 1 \pmod4$, then $$ \sum_{r=1}^{p-1}{(r|p) * r } \equiv 0 \pmod p.$$ ( $(r|p)$ is a Legendre Symbol ) I know that $\sum_{1 \le r \le p}{(\frac{r}{p})} = 0$, but ...
0
votes
1answer
45 views

Sum reminiscent of $(1+x)^N$ (binomial theorem)

I stumbled upon this sum while working on my thesis: $$\sum_{k=0}^N \binom{N}{2k} x^k$$ I know that $$\sum_{k=0}^N \binom{N}{k} x^k = (1+x)^N$$ But when it comes to the sum above I'm lost. Is ...