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

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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 \...
1
vote
2answers
36 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
104 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
232 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
90 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
145 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
46 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
37 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
35 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
29 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
46 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 ...
2
votes
1answer
68 views

Summation involving a hypergeometric 1F1 function

I'm trying to find a closed form for the following: \begin{equation} \sum_{n=0}^\infty \frac{(-1/4)_n}{n!(3/2)_n}\left(\frac{i}{2\tau}\right)^{n} {_1F_1(2n+1;2n+2;i k)} \end{equation} Using the ...
1
vote
5answers
72 views

Summation $ \sum_{k=-\infty}^n a^k$

Is there any formula that directly gives the result of this summation: $$ \sum_{k=-\infty}^n a^k$$ ?
0
votes
1answer
44 views

Kolmogorov's Truncation Lemma (iii)

Probability with Martingales: In the definition of $f$, is that really $z$ and not $\lceil |z| \rceil$, $\lfloor |z| \rfloor + 1$ or something? How exactly do we have the part in the $\...
1
vote
1answer
36 views

Modified Sum of Products

A given number k is to be expressed as a sum of products of integers keeping in mind that the integers used in above process do not exceed their cumulative sum as 100. For e.g., k = 19 can be ...
1
vote
2answers
41 views

Proving by induction that $\sum\limits_{i=1}^n\frac{1}{n+i}=\sum\limits_{i=1}^n\left(\frac1{2i-1}-\frac1{2i}\right)$

I have a homework problem to prove the following via induction: $$\sum_{i=1}^n \frac{1}{n+i} = \sum_{i=1}^n \left(\frac{1}{2i-1} - \frac{1}{2i}\right) $$ The base case is true. So far I've done the ...
1
vote
1answer
26 views

How to interpret summation convention?

In Landau and Lifschitz Mechanics, p. 99, we have (implicit) the equality $$\Omega_i^2 x_i^2 = \Omega_i \Omega_k \delta_{ik} x_{\ell}^2 $$ written with Einstein summation convention. The left hand ...
3
votes
1answer
41 views

Need hint on induction proof for summation

I have a homework problem to prove the following via induction: $$\sum_{i=1}^n i^22^{n-i} = 2^{n+3}-2^{n+1}-n^2-4n -6$$ The base case is true. I generated the below using $s_k+a_{k+1}=s_{k+1}$: $$ 2^{...
0
votes
1answer
40 views

Summing (and not summing) to $0\bmod9$

Let $n_0,n_1,n_2\in\{1,2,\dots,8\}$ and consider the sum \begin{align*} S&=\sum_{k=0}^8n_{k\bmod3}\\&=n_0+n_1+n_2+n_0+n_1+n_2+n_0+n_1+n_2. \end{align*} Is there an efficient way to ...
2
votes
5answers
50 views

Counting arguments Given one prove the other identity

Given: $${n \choose 0} + {n \choose 1} + {n \choose 2} + \cdots + {n \choose n} = 2^n$$ Prove the following in 2 ways. $$ {n \choose 1} + 2 {n \choose 2} + 3 {n \choose 3} + \cdots + n{n\choose n} =...
5
votes
8answers
110 views

Formula for sum of first $n$ odd integers

I'm self-studying Spivak's Calculus and I'm currently going through the pages and problems on induction. This is my first encounter with induction and I would like for someone more experienced than me ...
7
votes
1answer
82 views

Summation is to integration, as the large product operator is to…? [duplicate]

The integral is defined many ways but one that I am aware of is the Riemann Integral(?) which looks sorta like $\sum^n_{i=0} f(a +i\frac {b-a}n)*\frac {b-a} n$. An interesting thought is "is there a ...
3
votes
3answers
150 views

Is $\frac{d}{dx}\left(\sum_{n = 0}^\infty x^n\right) = \sum_{n = 0}^\infty\left(\frac{d}{dx} x^n \right)$ true?

Almost 3 months ago, I asked this question regarding if it's possible to compute the summation of derivatives, as in the example I've given: $$\sum_{n = 0}^\infty \frac{d}{dx} x^n$$ One answer ...
11
votes
4answers
607 views

Which is the explanation of the identity $\sum_{k=0}^n {n \choose k}k^2 = 2^{n-2}n(n+1)$?

A friend of mine put me a problem some time ago, and after trying to do it, I finally surrendered. I looked for the answer online so maybe I could guess why is it like it is, but I just can't ...
2
votes
1answer
46 views

Evaluation of Infinite series summation.

For any Positive integer $n\;,$ Let $t(n)$ denote the integer closest to $\sqrt{n}\;,$ Then value of $\displaystyle \sum^{\infty}_{n=1}\frac{2^{t(n)}+2^{-t(n)}}{2^n}$ $\bf{My\; Try::}$ Here ...
3
votes
2answers
67 views

How to find the summation of this infinite series: $\sum_{k=1}^{\infty} \frac{1}{(k+1)(k-1)!}(1 - \frac{2}{k})$

I've been trying to figure out the following sum for a while now: $$\sum_{k=1}^{\infty} \frac{1}{(k+1)(k-1)!}\left(1 - \frac{2}{k}\right)$$ I'm pretty sure that this doesn't evaluate to $0$. ...
0
votes
0answers
20 views

extracting a function out of an equation

I encountered the following problem in my thesis. We have an equation as follows: $\phi(s)=\sum^\infty_{n=1}P(n)\int^{\infty}_{0}e^{-st}f(t|n)dt=\sum^{\infty}_{n=1}[(1-q)M_1(s)]^{n-1}qM_2(s)$ in ...
0
votes
1answer
14 views

Value of the series $\sum_{n=1}^{\infty} H_n^{(2)}x^n$ and $\sum_{n=1}^{\infty} H_n^{(3)}x^n$

I can find series $$\sum_{n=1}^{\infty} H_nx^n = -\frac{\log(1-x)}{1-x}$$ but I can't find $$\sum_{n=1}^{\infty} H_n^{(2)}x^n$$ and $$\sum_{n=1}^{\infty} H_n^{(3)}x^n$$ $H_n^{(p)}=1+\frac{1}{2^p}+...
1
vote
0answers
24 views

Finding a closed mathematical form for a parametrized function series

I am dealing with the following parametrized function series defined by $$ F(x) := \sum_{n=0}^{\infty} 2\left( \varphi_n(x)-(n+1)(n+2)x^2 \psi_n(x) \right)x^{n+1} \, , $$ where $x \in [0,1)$. The ...
2
votes
1answer
70 views

Value of the series $\sum_{n=1}^{\infty}\big (\frac{H_n}{\binom{3n}{n}}\big)^2$

I want to find the value of following series $$ \sum_{n=1}^{\infty} \left(\frac{H_n}{\binom{3n}{n}}\right)^2\tag{1} $$ where $H_n = 1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n}$. I know that $$\sum_{...
6
votes
0answers
110 views

Help with the following summation when $x^{37}=1,x\neq 1$

I want to find the following summation $\text{Let }x^{37} = 1 \text{ and } x \neq 1,$ $\\ \text{Find the summation of }$ $$\frac{1}{(1+x+x^2+x^3)^3}+\frac{2}{(1+x^2+x^4+x^6)^3}+...+\frac{36}{(1+x^{...
1
vote
1answer
40 views

How sum of exponential variables is a gamma variable [duplicate]

I have the task to calculate $P(S_{100}\geq 200)$ where $S_{100}=\sum^{100}_{i=1} X_i$ and $X_i$, $i=1,2, \cdots, 100$ are independent $exp(\lambda)$ random variables. One method is to use the fact ...
2
votes
3answers
52 views

Summation of a series with 2 different methods gives 2 different answers

The objective here is to find the value of $S$, where $S$ is given by, $$S = 1-{1\over2}+{1\over3}-{1\over4}+...$$ I did this using two methods, but both the methods give different answers. Method 1: ...
3
votes
1answer
106 views

What should $\int \frac{1}{x} dx$ equal to?

Before you say that $\int \frac{1}{x} dx$ is equal to $\ln|x| +C$ due to positve and negative, I would like to show you why it is not convincing to me. Problem 1 and its possible solution. \begin{...
0
votes
1answer
28 views

a[i] denote number of friends i-th student has. c[j] denote frequency having at least j friends. Show that: ∑a[i]=∑c[j]. [duplicate]

Q. A class has 100 students. Let a[i], 1≤i≤100, denote the number of friends the i-th student has in the class. For each 0≤j≤99, let c[j] denote the number of students having at least j friends. Show ...
0
votes
5answers
431 views

Mathematical induction using Sigma [closed]

I have attached an image of a kind of mathematical induction question that i have never seen before. I attached it because i don't know how to type all the symbols out properly, i'm sorry again would ...
1
vote
1answer
46 views

Relation between $\gcd$ and Euler's totient function .

How to show that $$\gcd(a,b)=\sum_{k\mid a\text{ and }k\mid b}\varphi(k).$$ $\varphi$ is the Euler's totient function. I was trying to prove the number of homomorphisms from a cyclic group of order ...