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

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-4
votes
1answer
86 views

Definite Integral using its McLaurin Series.

I'm trying to solve the next integral, using its series. However, I got stuck in a very dumb way nearly at the end. The infamous: $$\int_{0}^{1} \frac{\text{cosh}(x)-1}{x}dx$$ First, the series of $\...
0
votes
2answers
101 views

Find the integer part of the sum $S=\sum_{k=1}^{80} \frac{1}{\sqrt k} $

Let $$S=\sum_{k=1}^{80} \frac{1}{\sqrt k}.$$Then I would like to obtain $\lfloor S \rfloor$, the integer part of $S$. I am not able to think how to start question .
6
votes
3answers
121 views

Compute $\sum\limits_{k=0}^{100}\frac{1}{(100-k)!(100+k)!}$

$$\sum_{k=0}^{100}\frac{1}{(100-k)!(100+k)!}$$ My work $$\sum_{k=0}^{100}\frac{2n!}{(2n!)(n-k)!(n+k)!}$$ $$\sum_{k=0}^{100}\frac{^{2n}C_{n-k}}{(2n!)}$$ $$\sum_{k=0}^{100}\frac{^{2n}C_{n+k}}{(2n!)}$$...
2
votes
0answers
31 views

Issues regarding my take on proving $E(X) = \lambda$, where $X\sim Poisson(\lambda)$

My proof: Let $X\sim \mathrm{Poisson}(\lambda)$. Then $$f_{\Tiny{X}}(x) = \frac{\lambda^x}{x!} e^{-\lambda}.$$ Thus, $E(X) = \sum_{x=0}^{\infty} x f_{\Tiny{X}}(x) = \sum_{x=0}^{\infty} x \frac{\lambda^...
27
votes
2answers
813 views

Proof or derivation of this identity $\lim_{n\to \infty}{\frac1{2^n}\sum_{k=0}^n\binom{n}{k}\frac{an+bk}{cn+dk}}\;\stackrel?=\;\frac{2a+b}{2c+d}$?

I just came up with the following identity while solving some combinatorial problem but not sure if it's correct. I've done some numerical computations and they coincide. $$\lim_{n\to \infty}{\frac{1}{...
0
votes
0answers
28 views

Show $\sum_{k=0}^n b_r(n,k) = (r-1)!\frac{x^{\bar{n}}}{(x+1)^{\bar{r-1}}}$ [duplicate]

Let's define $b_r(n,k)$ as $n$-permutations with $k$ cycles where numbers $1\dots r$ belong to one cycle. I tried to first define closed form for $b_r(n,k)$. My idea: We need to put $1 \dots r$ into ...
1
vote
1answer
52 views

How to get the last line?

I'm supposed to find the following equality: $\sum_{r=1}^{s-1}2^{2r-1} -\sum_{r=1}^{s-1}2^{r-1} +(n-2^{s-1}+1)(2^s-1)$ $=\frac{2}{3}(4^{s-1}-1)-(2^{s-1}-1)+(2^s-1)n-2^{2s-1}+3*2^{s-1}-1$ I ...
8
votes
2answers
71 views

Sum to closed form

I need to evaluate the following summation: $$ \sum_{n\in\mathbb{Z}} \frac{-1}{i(2n+1)\pi -\mu} $$ where $n$ is summed over all the integers from $-\infty$ to $\infty$ including 0. Putting this into ...
-6
votes
3answers
92 views

Is $\sum_{n=0}^\infty$ a misleading notation? [closed]

Why do we use the notation $\sum_{n=0}^\infty a_n$, for what is not defined as a sum, but is in fact a limit of a totally different expression? I understand what it means to sum a finite number of ...
1
vote
0answers
42 views

Does the following series converge to what I think it does?

I was wondering if the following series converges: $$\sum_{k=0}^\infty\frac{\Gamma(n+2)}{k!\Gamma(n+2-k)}B_kx^{-k}\tag1$$ And if so, does it converge to $$\frac{n+1}{x^{n+1}}\sum_{k=1}^xk^n\tag2$$ ...
4
votes
2answers
81 views

Find $\sum_{m=0}^n\ (-1)^m m^n {n \choose m}$

I'm going to university in October and thought I'd have a go at a few questions from one of their past papers. I have completed the majority of this question but I'm stuck on the very last part. In ...
1
vote
0answers
40 views

Interchanging sum and integral

If $\mathbf {i)} f_n\in C^1(\overline U)$ ($U$ is convex), $\mathbf {ii)} \sum\limits_{n\ge1}df_n$ converges unifromly on $U$ and $\mathbf {iii)} \sum\limits_{n\ge1}f_n$ converges in an arbitrary ...
4
votes
2answers
143 views

Evaluate $\lim_{n \to \infty }\left(\frac{1}{{n\sqrt{{n^2} + 1}}}+\frac{2}{{n\sqrt{{n^2}+4}}}+\cdots+\frac{n}{{n\sqrt{{n^2}+{n^2}}}}\right)$

Evaluate $$\mathop {\lim }\limits_{n \to \infty } \left(\frac{1}{{n\sqrt {{n^2} + 1} }} + \frac{2}{{n\sqrt {{n^2} + 4} }} + \frac{3}{{n\sqrt {{n^2} + 9} }} + \cdots + \frac{n}{{n\sqrt {{n^2} + {n^2}}...
2
votes
1answer
55 views

Distribution of distances between points with complete spatial randomness

I'm trying to compute the probability of the distances between points on a 2D domain that have complete spatial randomness (CSR). From this wikipedia page on CSR, the probability of locating the $N$...
2
votes
2answers
67 views

Prove by induction that $\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$

As the title says I need to prove the following by induction: $$\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$$ When trying to prove that P(n+1) is true if P(n) is, then I ...
5
votes
3answers
90 views

Factorial Proof by Induction Question? [duplicate]

$\text{Use the PMI to prove the following for all natural numbers n.}$ $ \frac{1}{2!} + \frac{2}{3!} + \cdot \cdot \cdot + \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!} $ So for this question I get ...
35
votes
9answers
2k views

Is difference of two consecutive sums of consecutive integers (of the same length) always square?

I am an amateur who has been pondering the following question. If there is a name for this or more information about anyone who has postulated this before, I would be interested about reading up on it....
0
votes
0answers
11 views

Summation over simplices in higher dimensions

Let $l\ge 1$ and $s\ge 1$ be integers. Define $\vec{a} := \left(a_j\right)_{j=0}^s$. The following set: \begin{equation} \Delta^{(s)}_l := \left\{ \vec{a} \quad \left| \quad \sum\limits_{\eta=0}^s a_\...
-5
votes
5answers
92 views

The 1000th partial sum of the series $\sum 1/n^2$ is less than 2 [closed]

Can anyone help me with this problem: Prove: $$1/1^2 + 1/2^2 +1/3^2 +\dots +1/1000^2 <2$$
4
votes
1answer
97 views

What is $\sum_{i=1}^{n}\frac{F_i}{i}$?

Mathematica is able to evaluate the summation $\sum_{i=1}^{n}\frac{F_i}{i}$ in terms of the Lerch transcendent. It is natural to consider whether or not this summation can be expressed in a more ...
2
votes
1answer
301 views

Reducing an indicator function summation into a simpler form.

Context I am attempting to reduce the space I need to store in an array in a program. I have made it so that the indices are always sorted. There are no indices where they are equal, and no indices ...
2
votes
1answer
57 views

factorial summation

I was trying to prove the problem here problem no 17 $$\displaystyle \sum_{n=0}^{\infty} \dfrac{2^{n-1} n!}{\prod_{r=0}^{n} (4r+1)} = \dfrac{\pi + 2\log(1+\sqrt{2})}{4 \sqrt{2}} $$ My Try: $$\...
1
vote
0answers
44 views

Solving recurrence relation $a_n=1 + \sum\limits_{i=1}^{n-1}ia_{n-i}$ with $a_1=1$

Consider the recurrence relation $$a_n=1 + \sum_{i=1}^{n-1}ia_{n-i}$$ with initial term $a_1=1$. What is $a_n$? I tried to guess some closed formula from the first 6 terms, which are $1$, $2$, $5$, $...
4
votes
1answer
66 views

Limit of a summation involving fractional parts

Working with some problems on the floor function, I noticed that the sum $$\frac {1}{n}\sum_{{\sqrt{n}}\leq x\leq n}\left\{\sqrt {x^2-n}\right\} $$ where $n$ and $x$ are integers, $\left\{f(x)\right\...
0
votes
1answer
31 views

The Fourier transform of the Bartlett (triangular) window

I am trying to understand how to obtain the Fourier transform of the Bartlett (triangular) window. The Bartlett window is defined as $$ w_B(k)=\begin{cases}\frac{N-|k|}N,& |k|\le N;\\0,&|k|>...
0
votes
1answer
34 views

sum of falling factorial $\sum_{k=0}^{n-1}\frac{n!}{(k+1)!}a^{n-k-1} $

I want to compute $\sum_{k=0}^{n-1}\frac{n!}{(k+1)!}a^{n-k-1}$. I note that it is similar to a generating function. The coefficients are falling factorials. Can I simplify it? Thanks!
0
votes
1answer
76 views

How was the integral for Zeta Function created

How was the zeta function integrated from $$\zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^{s}}$$ To $$\zeta(s) = \frac{1}{\Gamma (s)}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-1}dx$$ I've tried googling ...
2
votes
0answers
65 views

Identity involving the Catalan numbers and binomial coefficients

Let $C_k := \frac{1}{k + 1} \binom{2k}{k}$ be the $k$-th Catalan number and let $K$ be a positive integer. I am looking for an identity or simplification of \begin{equation} \sum_{k = 0}^K C_k \...
1
vote
1answer
55 views

Function $f: \mathbb{Z} \to \mathbb{Z}^n$ related to $\sum_{k=1}^{x} k^n$.

The sequence $\{a_0,a_1,...a_x\}$ has closed form $a_n=\sum_{i=0}^{\infty} \Delta^i(0) {n \choose i}$ where $\Delta a_n$ denotes the operation mapping $a_n$ to $a_{n+1}-a_n$ and $\Delta^i(0)$ is ...
1
vote
0answers
47 views

How can this double summation be solved?

I have to calculate the following expectation $$\mathbb{E}\left[\left(\frac1M\sum\limits_{i=1}^MX(i-n_1-M)\right)\left(\frac1M\sum\limits_{j=1}^MX(j-n_2-M)\right)\right]$$ where $M$, $n_1$ and $n_2$ ...
2
votes
1answer
94 views

Is there a closed formula for this summation?

I have the summation $$\sum\limits_{i=1}^n \frac{1}{i^2}$$ And I don't know how to find a closed formula for it. Any ideas?
2
votes
0answers
54 views

Fibonacci summation

Can anyone help me to prove the following relation. $$\sum_{k=1}^{\infty} \frac{F_{2k}H^{(2)}_{k-1}}{k^2\binom{2k}{k}}=\frac{2\pi^4}{375\sqrt{5}}$$ I was studying recently about Fibonacci and ...
0
votes
4answers
48 views

Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n - 1}$ using the binomial theorem

I'm trying to prove that \begin{equation} \sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n - 1} \end{equation} with the Binomial Theorem. I know that the B.T. states that \begin{equation} (x + y)^n = ...
0
votes
1answer
28 views

Pochhammer symbol finite summatory

I need some help in showing that in product among $n$ lower triangular matrices, the number of addends to be summed in order to obtain the value of the elements $(i, j)$ is: $\frac{<n>_{i-j}}{(i-...
0
votes
2answers
51 views

Genereating function of $H_{2n}$

We know the generating function of: $$\sum_{n=1}^{\infty}H_nx^n=\frac{\ln(1-x)}{x-1}$$. How do we find out the generating function of $$\sum_{n=1}^{\infty}H_{2n}x^n$$ I used the formula: $\...
1
vote
1answer
67 views

Riemann Zeta Function, Stirling's Numbers, and Infinite Series

A while back I was able to prove the following identity, $$\sum_{k=1}^{\infty}\frac{\Gamma(k+r)}{\Gamma(k)(k+r)^s}=\sum_{k=0}^{r}s(r+1,r+1-k)\zeta(s-r+k)$$ where $s(k,n)$ are the Sterling numbers of ...
3
votes
3answers
129 views

Has this summation a specific result?

I need to calculate this integral $$\sum\limits_{k=1}^\infty\int\limits_I \frac{\lambda^k}{(k-1)!}t_k^{k-1}e^{-\lambda t_k} dt_k$$ where $I=(a,b)$. Someone told me that summation equals $\lambda$, ...
0
votes
0answers
9 views

What should be expected values of weights in statistics (specifically a set of biweight weightings)

I am working on a project that involves taking the biweight sample variance of a velocity dataset. This is defined as $\sigma_{BI}^2 = N\dfrac{\sum_{|u_i|<1}(1-u_i^2)^4(v_i-\bar{v})}{D(D-1)}$ ...
-1
votes
1answer
37 views

Reducing Binomial Summation [closed]

How can I reduce this summation into this $$\frac{1}{2}(1+\left(1/3\right)^{50})$$ The problem comes from the 1992 AHSME Test (problem 29)
0
votes
1answer
23 views

Convergence of $\sum_{n\in\mathbb{N}_{>0}} f(n)\mathrm{log}(n)$

A function $f(n)$ has the following conditions: $$ f(n),n\in\mathbb{N}_{>0} $$ $$ f(n)\in[0,1] $$ $$ \sum_{n\in\mathbb{N}_{>0}} f(n)=1 $$ Does the following sum always converge? Or does a ...
5
votes
2answers
79 views

How to find $\sum_{A \subset S} (\min A)$ and $\sum_{A \subset S} (\max A)$ if $S=\{1,2,…,n\}$?

Here, $\min A$ and $\max A$ denote the minimum and maximum element respectively of the set $A$. So I have to calculate how many subsets of S have min/max element $1$, how many subsets have min/max ...
4
votes
3answers
68 views

Pattern with the the tetration of summations.

While dealing with a question with finding an explicit form for a sequence I noticed something: $$\sum_{x_0=0}^{n-1} 1=\frac{n}{1!}$$ $$\sum_{x_0=0}^{n-1} \sum_{x_1=0}^{x_0-1} 1=\frac{n(n-1)}{2!}$$ ...
0
votes
1answer
35 views

If $(A_n)_{n\in \mathbb{N}}$ is an open set in $\mathbb{R}^2$ how do I find an example where $\bigcap\limits_{n=1}^\infty A_n$ also is open?

If $(A_n)$ is an open set in $\mathbb{R}^2$ for all $n\in \mathbb{N}$. How can I find an example where $\bigcap\limits_{n=1}^\infty A_n$ also is open? I'm new to this concept, so any leads would be ...
3
votes
1answer
43 views

logarithm of a sum or addition

I search a general rule for calculating the logarithm of a sum or addition. I know that $$\ln{(a+b)}=\ln{\left(a\left(1+\frac b a\right)\right)}=\ln{(a)}+\ln{\left(1+\frac b a\right)}$$ but when the ...
0
votes
3answers
56 views

How to calculate $x=\sum _{ i=0 }^{ \infty } \left( i+1 \right) \cdot \left( \frac { 5 }{ 6 } \right) ^{ i }$ [duplicate]

I was trying to find the value of x in the following equation. $$x=\sum _{ i=0 }^{ \infty } \left( i+1 \right) \cdot \left( \frac { 5 }{ 6 } \right) ^{ i }$$ In a computer simulation, I found that ...
0
votes
2answers
30 views

What's wrong with this formula for the dot product of a vector and a matrix acting on that vector?

Suppose we have an $n \times n$ matrix $M$ and a vector $v$. I want to find an explicit formula for $v \cdot Mv$. I begin by saying $$v \cdot Mv = \sum_{i=1}^{n} v_i(Mv)_i$$ and since $Mv$ is given by ...
5
votes
1answer
66 views

Sum of nth powers of Fibonacci numbers

Is a closed form for $$\sum_{i=1}^n{F_i^k}$$ (where $F_i$ is the $i^{th}$ Fibonacci number and $k$ is constant) known?
0
votes
0answers
31 views

Can I calculate a fractional sum with functional equations and/or infinite series?

I was wondering if I can calculate fractional sums (non-integer sums) with functional equations in the following manner $$f(x):=\sum_{n=1}^xg(n)$$ $$f(x)=g(x)+f(x-1)\tag1$$ $(1)$ most certainly ...
0
votes
0answers
25 views

“Solving” for a sequence given an (expected) expression for the summation

Consider the "equation" \begin{equation} \frac{1}{a_n}\sum_{k=1}^n ka_k = \mathcal{O}\left(\frac{n^2}{\log n}\right).\tag{1}\label{eq:conjec} \end{equation} Does there exist some monotonically ...
1
vote
2answers
29 views

Summation Notation Question in McMillan's Theorem Proof

Let me preface by saying that this question does not pertain as much to coding theory, as it does to mathematical notation. Every symbol in this question is a natural number. Anyhow, I am currently ...