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

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0
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1answer
14 views

“Sum of power” for prime numbers

I use Euler–Maclaurin formula, Faulhaber's formula and Bernoulli polynomials for "sum of powers" for this type $\sum_{t=1}^nt^k$. but I don't know to find compact form when sum is taken from first ...
0
votes
0answers
13 views

What is the difference between finding the sum of a series and its closed-form solution?

In complexity theory, it is sometimes necessary to find the "closed-form solution" of a summation. This was put in our exam guide as "solving arithmetic and geometric series", which I initially ...
1
vote
0answers
17 views

Find $Z$ transform of given signal

Given the discrete signal $h(n)=r^n\frac{\sin{[(n+1)\theta]}}{\sin{\theta}}$ if $n \geq 0$ and $h(n)=0$ otherwise, find the $Z$ transform of $h(n)$. What I did: We know that ...
2
votes
1answer
50 views

Looking for formula of $\sum_{k=1}^m (-1)^k \dfrac {x^2(x^2-1)…(x^2-k+1)}{(x+1)(x+2)…(x+k)}$

Let \begin{equation*} u_k:=(-1)^k \dfrac {x^2(x^2-1)...(x^2-k+1)}{(x+1)(x+2)...(x+k)}. \end{equation*} Can we find the sum of first $m$ of $u_k$ 's? That is, is there any formula for $\sum _{k=1}^m ...
3
votes
1answer
53 views

Other variation of Nicomachus's Theorem?

We all know that $ 1^3+2^3+3^3 + \ldots + n^3 = (1+2+3+\ldots + n)^2 $. Denote $\displaystyle S_m = \sum_{j=1}^n j^m $, then we can set $ S_3 = S_1 ^2 $ for all positive integers $ n $. Question: Is ...
6
votes
1answer
269 views
+50

Integral of the ratio of two exponential sums

I am trying to find a lower bound on the following integral \begin{align*} \int_{y=-\infty}^{y=\infty} \frac{ (\sum_{n=[-N..N]/\{0\}}n e^{-\frac{(y-cn)^2}{2}})^2} {\sum_{n=[-N..N]/\{0\}} ...
1
vote
1answer
47 views

How to prove that Grandi's series $= \frac{1}{2}$ using Euler transform [closed]

Let $x$ denote Grandi's series $1-1+1-1+1-1+1-...$ This implies that $$ x = 1\text{ or}\\ x = 0\text{ or}\\ 1-x = 1 - (1-1+1-1+1-...) = x \implies 2x = 1 \implies x = \frac{1}{2}$$ Where the last ...
0
votes
4answers
70 views

Prove that $\sum\limits_{r \mathop= 0}^n \frac {(-1)^r} {r!(n-r)!} = 0$

I wish to prove that $\displaystyle \sum_{r \mathop= 0}^n \frac {(-1)^r} {r!(n-r)!} = 0$ It is plain when $n$ is an odd integer. How might one go about proving it generally?
0
votes
0answers
14 views

Estimation for a logarithmic function in $(0,\,1)$. A series should be used?

Let $f(t)\geq C_1t^{-\alpha}$ for all $t\in(0,\,\infty)$ and for some $C_1>0,\,\alpha>0$. and let $g(t)\geq C_2\left(\ln(t^{-1})\right)^\beta$ for all $t\in(0,\,1)$ and for some ...
3
votes
0answers
40 views

Elemenatry topological proof of Erdos conjecture on Arithmetic Sequences

Define $C_n(A) = \{a \in A : \forall d \in \Bbb{N}$ one of $a + d, a +2d, \dots a + (n-1)d$ is not in $A\}$ For $n \leq m$, we have $C_n(A) \subset C_m(A)$. Proof. Let $x$ be in the LHS. Then ...
0
votes
1answer
49 views

Help with sequence $1\cdot n + 2(n-1) + \ldots + (n-1)2 + n\cdot 1$

Can anyone please provide a simplified formula for the sum of the sequence \begin{equation*} s(n) = 1\cdot n + 2(n-1) + \ldots + (n-1)2 + n\cdot1 \end{equation*} where $n$ is an integer greater than ...
1
vote
0answers
12 views

Abel's summation formula for functions depending on limit of sum

Abel's summation formula states that for two functions $f$ and $g$, with $f$ differentiable, we have $$\sum_{k=1}^n f(k)g(k)=G(n)f(n)-\int_1^n G(x)f'(x)\; dx \tag{$*$}$$ where $G(n)=\sum_{k=1}^n g(k)$ ...
0
votes
1answer
17 views

Proof of Lagrange Identity

I need to prove Lagrange Identity for complex case, i.e. $$ \left( \sum_{i=1}^n|a_i|^2 \right)\left( \sum_{i=1}^n |b_i|^2 \right)-\left| \sum_{i=1}^na_ib_i \right|^2=\sum_{1\leq i<j\leq ...
1
vote
7answers
183 views

How to show $\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$?

Show that $\,\displaystyle\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$. I'm thinking right now (though not getting anywhere with it) that I want to expand out the summation portion to ...
5
votes
2answers
176 views

Finite Summation of Fractional Factorial Series

Is there a closed form solution for the following series? (Without Using Gamma Function): $$ S=\sum _{i=1}^{n-1} \frac{1}{(i+1)!} $$
-2
votes
3answers
80 views

Understand the steps in a Summation [closed]

I don't understand the following steps: \begin{align*} \sum_\limits{t=1}^{n-2} (-1)^{t-1} {n \choose t}a\cdot\tfrac{1}{2}\cdot(n - t)(n - t - 1) & = a\cdot\tfrac{1}{2}\cdot n(n - 1) ...
3
votes
1answer
37 views

Prove Lagrange's Identity without induction

Prove Lagrange's Identity without induction. $$ \sum_{1\leq j <k\leq n}(a_jb_k-a_kb_j)^2=\left( \sum_{k=1}^na_k^2 \right)\left( \sum_{k=1}^n b_k^2 \right)-\left( \sum_{k=1}^na_kb_k \right)^2 $$ I ...
4
votes
0answers
32 views

Questionable Convergence of a Series

The summation is: $$ S = \sum_{k \geq 0} f(k) \int_{0}^{\pi/2} \sqrt{1-(1- \frac{f(k+1)^2}{f(k)^2})\sin^2(\theta)}d\theta $$ Now, we know that $f(k+1) < f(k)$ and as $k$ approaches infinity, ...
2
votes
1answer
47 views

Average of elements in a subset of $\{1,2,3,..,n\}$ is greater than $\frac{n+1}{2}$ [on hold]

Consider two integers $n \ge m \ge 4$ and $A=\{a_1,a_2,...,a_m\}$ a subset of the set $\{1,2,3,...,n\}$ with the property that $$\forall a,b \in A \text{ with } a \neq b, \text{ if } a+b \le n, \text{ ...
0
votes
4answers
35 views

How do I read this triple summation? $\sum_{1\leq i < j < k \leq 4}a_{ijk}$

How do I read this triple summation? $$\sum_{1\leq i < j < k \leq 4}a_{ijk}$$ The exercise asks me to express it as three sumations and to expand them in the following way: 1) Summing first ...
0
votes
2answers
32 views

Closed form formula for discrete sums [on hold]

Is there a general way to obtain a closed form formula for any discrete sum of the form: $\sum_{a}^{b}f(n)$ with certain restrictions on the form of $f(n)$, much like how we can find closed form ...
5
votes
5answers
615 views

Given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$

I tried to solve it using induction, but that got me no were, in the middle of the equation stat appearing ks that I don't see how to get out of the equation. I think the easiest way to prove it, it's ...
0
votes
0answers
26 views

Sum of products of K numbers taken from N numbers in closed form

Let's say i have 5 numbers, $A,B,C,D,E$. I want to know the sum of all the possible products of some or all of these numbers each taken at most once. Instead of a lot of multiplications and additions ...
12
votes
2answers
1k views

Floor function properties: $[2x] = [x] + [ x + \frac12 ]$ and $[nx] = \sum_{k = 0}^{n - 1} [ x + \frac{k}{n} ] $

I'm doing some exercises on Apostol's calculus, on the floor function. Now, he doesn't give an explicit definition of $[x]$, so I'm going with this one: DEFINITION Given $x\in \Bbb R$, the ...
4
votes
3answers
126 views

Show that if $x\geq 0$ and $n$ is a postive integer, then $\sum_{k=0}^{n-1}\left\lfloor {x+\frac{k}{n}}\right\rfloor=\lfloor {nx}\rfloor$ [duplicate]

I need help with this question, where $\lfloor x\rfloor$ means the floor function of $x$. Show that if $x\geq 0$ and $n$ is a postive integer, then $$\sum_{k=0}^{n-1}\left\lfloor x+\frac{k}{n} ...
13
votes
6answers
190 views

show that $\frac{1}{F_{1}}+\frac{2}{F_{2}}+\cdots+\frac{n}{F_{n}}<13$

Let $F_{n}$ is Fibonacci number,ie.($F_{n}=F_{n-1}+F_{n-2},F_{1}=F_{2}=1$) show that $$\dfrac{1}{F_{1}}+\dfrac{2}{F_{2}}+\cdots+\dfrac{n}{F_{n}}<13$$ if we use Closed-form expression ...
0
votes
1answer
1k views

Find an expression for the area under the graph of f(x) as a limit?

$f(x) = \frac{2x}{x^2 +1}, 1 \leq x \leq 3$ Basically, I need to find an expression for the area under the graph within these intervals for the function as a limit. I understand the concept of the ...
6
votes
2answers
243 views

Finite Series $\sum_{k=1}^{n-1}\frac1{1-\cos(\frac{2k\pi}{n})}$

I want to show that $$\sum_{k=1}^{n-1}\frac1{1-\cos(\frac{2k\pi}{n})} = \frac{n^2-1}6$$ With induction I don't know how I could come back from $\frac{1}{1-\cos(\frac{2k\pi}{n+1})}$ to ...
0
votes
2answers
40 views

Calculus Summations

Is there any way I can change a summation, say from $k=1$ to $n$ of the derivative of order $k$ of a function into closed form, or some form that would be more manageable? Ex. From $k=1$ to $3$ of ...
0
votes
1answer
25 views

Is it possible to exhibit a collection of sets

Let a subset $D$ of the natural numbers be called convergent or divergent when the associated series $\sum_{d \in D} \frac{1}{d}$ converges or diverges. Define a topology on $\Bbb{N}$ by defining the ...
1
vote
0answers
52 views

Do sum indices need to be integers?

I have this exercise: Compare if these two sums are equal: $\sum_{k=0}^5a_k$ and $\sum_{k^2=0}^5a_{k^2}$ I know the first one is $a_0+a_1+\cdots+a_5$, but I'm wondering if the second one is the ...
4
votes
1answer
97 views

A convolution involving binomials

Given $$f(i)\gt0,\:g(i)>0,\:i =0,1,2,3,...\:$$and$$\sum_{i=0}^{\infty}f(i) = 1,\sum_{i=0}^{\infty}g(i) = 1$$Prove that, if$$\frac{g(l-k)f(k)}{\sum_{i=0}^{l}f(i)g(l-i)}=\binom{l}{k}p^k(1-p)^{l-k}\: ...
2
votes
2answers
51 views

Sum of the series $\tan^{-1}\frac{4}{4n^2+3}$

Find the value of $$\sum^{n=k}_{n=1}\tan^{-1}\frac{4}{4n^2+3}$$ I tried multiplying numerator and denominator by $n^2$, but got nothing. How do I split the term inside $\tan^{-1}$?
0
votes
1answer
39 views

Proof for Mean of Geometric Distribution

I am studying the proof for the mean of the Geometric Distribution http://www.math.uah.edu/stat/bernoulli/Geometric.html (The first arrow on Point No. 8 on the first page). It seems to be an arithco ...
1
vote
1answer
42 views

Strong Induction Proof

Prove that $$\sum_{j=1}^n (j)(j+1)(j+2)\cdots(j+k-1) = \frac{n(n+1)(n+2)\cdots(n+k)}{k+1}$$ Hint: $P(n, k)$ is true for all pairs of positive integers $n$ and $k$ if: (a) $P(1, 1)$ is true and $P(n ...
1
vote
2answers
37 views

Is there any way to find sum of sequence generated by formula?

I have such sequence: $2^2$, $7^2$, $12^2$, $17^2$, $22^2$, ... I found a formula to generate n-th term: $(5\cdot n+2)^2$ And I need to find a sum of those numbers: 4, 49, 144, 289, 484, ... Can ...
1
vote
0answers
33 views

Fibonacci Sequence (conceptual/mechanical)

I'm in Calc II and have my exam on series/sequences tomorrow. There will be an extra credit question related to Fibonacci and am trying to gather as much info on the sequence as possible. -If the ...
0
votes
0answers
32 views

series and dyadics [duplicate]

I'm in Calc II and am unfamiliar with what dyadics is but my teacher said that it's possible to find the sum of $\frac{\cos n}{n^2}$ by using dyadics. Would you mind laying out a step by step? ...
0
votes
1answer
22 views

Express sum with variable number of terms

I have the following data for a sum with variable $n$. For $n=1$, $\text{sum}=p^2+q^2$ For $n=2$, $\text{sum}=3p^2q+q^3$ For $n=3$, $\text{sum}=p^4+6p^2q^2 +q^4$ For $n=4$, ...
4
votes
3answers
105 views

Prove by induction that $1+4+7+…+(3n-2) = 2n(3n-1)$

I have an exercise where I, using induction, have to prove the following: \begin{equation*} 1 + 4 + 7 + \ldots + (3n-2) = 2n(3n-1). \end{equation*} I immediately got stuck on the base case with ...
0
votes
1answer
56 views

Does the infinite sum of $\sin(\frac{1}{n}+n\pi)$ converge or diverge? [on hold]

Does the infinite sum of $\sin(\frac{1}{n}+n\pi)$ converge or diverge? I would like to know which method is used to test this infinite series.
0
votes
1answer
14 views

Confused about a step with the summation operator.

Image 1 contains the step which I am confused about. What happens to the middle term (-XiY(bar) -YiX(bar)). Picture 2 contains the question for context. I understand how to do the question once I ...
0
votes
0answers
36 views

Solving a ratio of summation

I have to solve the equation \begin{equation*} y = \frac{\sum_{j=1}^m a_j x'_j}{\sum_{j=1}^m a_j x_j} \end{equation*} We have $\sum_{j=1}^m a_j \frac{x'_j}{x_j} = 1$ and $\sum_{j=1}^m a_j = 1$ ...
0
votes
3answers
29 views

Solution check: Let $f_n$ be fibonacci numbers. Prove: $\sum_{k=0}^{n-1} \binom{n+k}{2k+1} = f_{2n-1}$ and $\sum_{k=0}^n \binom{n+k}{2k} = f_{2n}$

The question: Let $f_n$ be fibonacci numbers. Prove: $\sum_{k=0}^{n-1} \binom{n+k}{2k+1} = f_{2n-1}$ and $\sum_{k=0}^n \binom{n+k}{2k} = f_{2n}$ For every $n\in N$. $f_0=f_1=1$, ...
1
vote
1answer
37 views

Interpretation of summations in regards to combinatorics

I've been studying for a final in combinatorics and ran into 3 different summations that have me stumped. 1) interpret the equation in terms of counting words. (Hint: $e^a$$e^b$$e^c$) $$e^{3x} = ...
0
votes
2answers
29 views

Summation problem: $\sum_{k=1}^6 (k-1)/(k+1)$

How do I solve: $\sum_{k=1}^6 \frac{k-1}{k+1}$ Thanks
2
votes
1answer
24 views

Uncountable sums convergence doubt

Hi I am having certain doubts about the sum of uncountable numbers In my class on functional analysis we proved that if $\sum_{\alpha \in A} X_{\alpha}$ converges then $X_{\alpha}$ is not equal to ...
3
votes
3answers
126 views

Find $S=\sum_{n=-\infty}^{\infty} \frac{(-1)^n}{1+n^2}$

How do I find the sum: $$S=\sum_{n=-\infty}^{\infty} \dfrac{(-1)^n}{1+n^2}$$ I can't solve this can someone help me?
-1
votes
0answers
19 views

Closed form for $\sum _{k=r}^s \binom{n}{k}$

The cardinality of the Powerset is $2^n$. Looking for $\sum _{k=r}^s \binom{n}{k}$, Mathematica gives $$\sum _{k=r}^s \binom{n}{k}=\binom{n}{r} \, _2F_1(1,r-n;r+1;-1)-\binom{n}{s+1} \, ...
1
vote
0answers
31 views

Can summation $\sum_{n=[-N\ldots N]}n e^{-\frac{(y-cn)^2}{2}}$ be lower bouned by integration $\int_{-N}^Nx e^{-\frac{(y-cx)^2}{2}} dx $

I was wondering if the following summation \begin{align*} \sum_{n=[-N \ldots N]}n e^{-\frac{(y-cn)^2}{2}} \end{align*} can be lower bounded by integral $$ \int_{-N}^Nx e^{-\frac{(y-cx)^2}{2}} \, dx ...