# Tagged Questions

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### 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 ...
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### 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.
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### $\sum _{j=0}^{\infty }\binom{-p-1}{j} \bigl( -\frac {x} {1+x}\bigr) ^{j}=?$

I did try to use geometric series somehow. I have no idea how to evaluate in terms of $p$ and $x$.
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### 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 ...
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### 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 $\delta(t-t_0)$. ...
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Pair of fair die are rolled (independently I hope) infinitely. Find probability sum of 5 appears before sum of 7. 2 approaches: $$P(\text{sum of 5 appears before sum of 7})$$ $$= P(\text{roll 1 ... 0answers 19 views ### summation involving a hypergeometric 2F2 function im trying to find the closed form for the following \sum_{n=0}^\infty \frac{c^n}{n!}\frac{(a)_n}{(b)_n}\frac{(\alpha+1/2)_n}{(\alpha+3/2)_n}{_2F_2}(-n,1-b-n;1-a-n,1/2;-\frac{d}{c}) \... 2answers 66 views ### Theorem 3.16. in Analytic Number Theory by Apostol The below texts are from the book Introduction to Analytic Number Theory by Apostol: I have two questions which I couldn't find solutions for them: 1- According to Thm 3.16., \sum_{n\le x} \... 0answers 36 views ### Convert Gaussian sum to integral? Consider a sum of the following form:$$S=\sum_{n=0}^\infty e^{A-B(n-C)^2}$$with constant A,B,C>0. Is there any way to convert this sum to an integral and evaluate via Gauss integral? Maybe it ... 2answers 263 views ### Evaluate  \int_{0}^{1} \log\left(\frac{x^2-2x-4}{x^2+2x-4}\right) \frac{\mathrm{d}x}{\sqrt{1-x^2}}  Evaluate :$$ \int_{0}^{1} \log\left(\dfrac{x^2-2x-4}{x^2+2x-4}\right) \dfrac{\mathrm{d}x}{\sqrt{1-x^2}} $$Introduction : I have a friend on another math platform who proposed a ... 1answer 57 views ### A sum of squared binomial coefficients I've been wondering how to work out the compact form of the following.$$\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 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 ... 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})$$I have to find the limit of this sequence , I tried to surround it ,but this not the correct way. 1answer 65 views ### Summing power series \sum_{n=1}^\infty \frac{(2n-1)!!}{(2n)!!}x^n Lets have series$$\sum_{n=1}^\infty \frac{(2n-1)!!}{(2n)!!}x^n$$Obviously, its convergence radius is 1. I should sum it, but don't know what's up with the double factorial. There is a hint in the ... 0answers 21 views ### Does anyone know a summation formula for Wilson's Polynomials? Wilson's polynomials are defined as W_n(x^2; a, b, c, d) := (a+b)_n (a+c)_n (a+d)_n {\space}_4F_3(-n, n+a+b+c+d-1, a+ix, a-ix;a+b, a+c, a+d; 1)  Does anyone know a summation formula for Wilson's ... 0answers 14 views ### Infinite exponential sum doubt Hello! I have a couple of doubts regarding a formula seen here :$$\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)}}$$For p,q\in\mathbb{R}. Obviously, ... 1answer 113 views ### How to justify interchange of sum and integral in fourier series? f is the 4-periodic function f(x) = 1 if x \in [0,2) and f(x) = - 1 if x \in [2,4) The Fourier series of f is$$F(t) = {4 \over \pi} \sum_{n=1}^{\infty} {\sin({\pi \over 2}(2n + 1)t) \...
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 ...
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 $L \... 1answer 13 views ### summation notation of elements in several different (sub)sets If there are two different sets$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 ... 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 ... 0answers 75 views ### Verify$y=x^{1/2}Z_{1/3}\left(2x^{3/2}\right)$is a solution to$y^{\prime\prime}+9xy=0This 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 $$x^... 1answer 120 views ### Intuitive ways to get formula of binomial-like sum Is there an intuitive way, though I am not sure how to find a conceptual proof either, to establish the following identity:$$\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$$for all natural ... 3answers 134 views ### Calculate \int_0^{1/10}\sum_{k=0}^9 \frac{1}{\sqrt{1+(x+\frac{k}{10})^2}}dx How can we evaluate the following integral:$$\int_0^{1/10}\sum_{k=0}^9 \frac{1}{\sqrt{1+(x+\frac{k}{10})^2}}dx$$I know basically how to calculate by using the substitution x=\tan{\theta}... 3answers 65 views ### Infinite Sum of Falling Factorial and Power According to Mathematica,$$\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)!}$$is the falling factorial. I would ... 4answers 137 views ### Prove if n \in \mathbb N, then \frac{1}{2!}+\cdots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!} Prove if n \in \mathbb N, then \frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\cdots+\frac{n}{\left(n+1\right)!} = 1-\frac{1}{\left(n+1\right)!} So I proved the base case where n=1 and got \frac{1}{2}... 2answers 76 views ### does anyone know a nice form of the infinite sum \sum_{n=0, m=0}^{\infty} \frac{a^n b^m}{(n+m)!}? I was wondering if anyone on here knows of a closed form or special function for this infinite sum:$$\sum_{n=0, m=0}^{\infty} \frac{a^n b^m}{(n+m)!}$$Or the sum of any non-trivial subset. 1answer 46 views ### Series expansion of {x} Hello and sorry for my bad English. I am not mathematician, so sorry if this seems a silly question. I've seen this formula regarding the fractional part of a number in Wikipedia, and I would like to ... 0answers 21 views ### Check if the sum is equal to the polynomial I have the following polynomial$$(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$$I believe I can ... 2answers 99 views ### Show that \sum_{r=1}^n r^4=\frac{3n^2+3n-1}5\sum_{r=1}^n r^2 Following from the question here, I was wondering if it's possible show directly that$$\sum_{r=1}^n r^4=\frac{3n^2+3n-1}5\sum_{r=1}^n r^2$$without expanding the summation in full on either side. 0answers 26 views ### How to get analytical summation of this series? How to get the analytical summation of this series?$$\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}}}} = ?$$... 1answer 31 views ### Evaluating a series with a constant as a bound I'm trying to find an expression to evaluate a series given a bound that is some unknown constant. For example, the simple summation below can be expressed as such: \sum \limits_{x=1}^{n}x=\frac{n(... 1answer 62 views ### Simplification of a double summation in a polynomial ring over a finite field I am looking out to simplify the following double summation in \mathbb{F}_q[x_1,x_2], where p is a prime and q=p^k for some positive integer k and a positive integer r such that 0 \leq r \... 2answers 35 views ### I need help reindexing the sum I know this is probably exceedingly simple, but I'm just stuck and keep making some mistake. Here, t_n represents the n-th Tribonacci number. That is, t_0 = 0, t_1 = 0, t_2 = 1 and t_n = t_{n-1}... 5answers 102 views ### Sum of combinatorics sequence \binom{n}{1} + \binom{n}{3} +\cdots+ \binom{n}{n-1} I need to find sum like$$\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}=\ ?$$3answers 213 views ### proof of 1^4+2^4+…+n^4=\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30} [duplicate] I want a 'simple' proof to show that:$$1^4+2^4+...+n^4=\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}$$I tried to prove it like the others but I can't and now I really need the proof. Also I want a geometric ... 2answers 88 views ### Determine \frac{f''(\frac{1}{2})}{f'(\frac{1}{2})} if f(x) = \sum_{k=0}^{1000} \ {2015 \choose k}\ x^k(1-x)^{2015-k} Problem : Determine \frac{f''(\frac{1}{2})}{f'(\frac{1}{2})} if f(x) = \sum_{k=0}^{1000} \ {2015 \choose k}\ x^k(1-x)^{2015-k} Trying to simply brute force the problem, yields the following ... 3answers 38 views ### Finite sub-sums of finite, countably infinite sums. Suppose that I is a countable set and$$ \sum_{i \in I} X_i = X,$$where X \in \mathbb{R} (in particular X is finite). Does this mean that for all \epsilon >0 there exists a finite subset ... 2answers 54 views ### Finding convergence zone/range for \sum_{i=1}^\infty \frac{x^{n^2}}{n(n+1)}$$\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}$$What steps do I need to take to continue? Looking for hints or steps, not full solution/ 3answers 143 views ### Does this \lim_{n \to +\infty} \frac{1}{n^2} \sum_{k=1}^{n} k \ln\left( \frac{k^2+n^2}{n^2}\right ) exist? I need to examine whether the following limit exists, or not.$$\lim_{n \to +\infty} \frac{1}{n^2} \sum_{k=1}^{n} k \ln\left( \frac{k^2+n^2}{n^2}\right )$$If it does, I need to calculate its value. ... 2answers 45 views ### Recurrence Relation with two parameters and Summation This is a recurrence relation with two parameters which came up in a problem I was trying to solve. Given$$\begin{align}&A_n=pB_{n-1};\qquad &&B_n=q(A_{n-1}+B_{n-1})\\ &A_4=p; \... 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'... 3answers 33 views ### Proof thatQo(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$? 1answer 27 views ### Summation Closed form for floor$\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 ... 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 ... 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 ...