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

learn more… | top users | synonyms

3
votes
3answers
57 views

Bounding $\sum_{k=1}^N \frac{1}{1-\frac{1}{2^k}}$

I'm looking for a bound depending on $N$ of $\displaystyle \sum_{k=1}^N \frac{1}{1-\frac{1}{2^k}}$. The following holds $\displaystyle \sum_{k=1}^N \frac{1}{1-\frac{1}{2^k}} = \sum_{k=1}^N ...
0
votes
1answer
54 views

Why does $ \sum_{i=0}^n \theta_i X_i = \theta^TX$ hold? [closed]

While reading Machine learning, I came through a formula, which is $$ \sum_{i=0}^n \theta_i X_i = \theta^TX .$$ I would like to know the name of the equation and some explanation behind it.
0
votes
3answers
42 views

summation for $x \gt 1$

how do I show that the $\sum_{n=0}^{n=N} (n+1)x^n$ is less than or equal to the square of this whole sum : $\sum_{n=0}^{n=N}x^n$ Tried induction didn't work. Got messy tried doing directly by using ...
1
vote
0answers
28 views

Simplifying Repeated Summation

Is it possible to simplify the expression below in order to reduce it to a single sum? Would Pascal's Triangle possibly be applicable to finding a solution? $$ \sum_{a_1=1}^n \sum_{a_2=1}^{a_1} ...
2
votes
3answers
97 views

Proving $\int_0^n \left(1-\frac{t}{n}\right)^n\ln(1/t)\,dt \to \gamma$

I have to prove that $\displaystyle \lim_{n\to\infty}\int_0^n \left(1-\frac{t}{n}\right)^n\ln(1/t)\,dt= \gamma$ I tried to expand $\left(1-\frac{t}{n}\right)^n$ and swap sum and integral, which ...
1
vote
5answers
96 views

How to calculate $\sum_{n=1}^\infty \frac{2}{(2n-1)(2n+1)}$ [closed]

How can I calculate the sum of the following infinite series: $$\sum_{n=1}^\infty \frac{2}{(2n-1)(2n+1)}$$
2
votes
0answers
13 views

A geometric-like sum with N variables

I want to compute the following sum $F_N=\sum_{1\le n_1<n_2\cdots<n_N\le L} a_1^{n_1} a_2^{n_2}\cdots a_N^{n_N}$ for rather large $L$. Is there a general formula for this kind of sum ?
0
votes
0answers
15 views

I attempt to combine Abel's summation with Hardy's inequality

Let $a(n)$ be a sequence of real numbers, $A(x)=\sum_{n\le x}a(n)$, with $A(x)=0$ if $x<1$ and $G(x)=\int_0^x g(t)dt$, with $g(t)\ge 0$ integrable on $[0, \infty)$, $p>1$ (is a requeriment for ...
0
votes
1answer
55 views

why doesn't proof of sum of two rational number is rational not proving the irreducibility of fraction $\frac{ad+bc}{bd}$?

When I was comparing proof for $\sqrt{2}$ and sum of two rational numbers, I found that the proof of two rational number did not mention anything about common factor in the ratio. one proof I found ...
3
votes
7answers
110 views

How do you solve the summation of $2-4+8-16+32- \dots 2^{48}$?

This is a summation problem but I can't seem to figure out how to solve this with the mix of subtraction and addition.
0
votes
0answers
31 views

How to calculate sum of sum of subsets using polynomial multiplication?

Recently I was reading about polynomial multiplication, and came across solution of one interesting problem. Which is finding sum of product of all $k-subsets$. which is as following: if we have ...
0
votes
1answer
32 views

Linearity of the supremum

Short question: In which cases we have: Let $f_n$ be a sequence of functions, then $\sup_{k\in \mathbb N}(\sum_{n}^if_k(n))=\sum_{n}^i\sup_{k\in \mathbb N}f_k(n)$ ? I guess if the $f_k(n)$ are ...
0
votes
5answers
88 views

$\sum_{i=0}^n {2n \choose 2i} = 2^{2n-1}$

$$ \sum_{i=0}^n {2n \choose 2i} = 2^{2n-1} $$ I know what this sum is supposed to equal. I also have a hint that I am supposed to use ${n \choose r} = {n-1 \choose r-1} + {n-1 \choose r}$ I was ...
0
votes
0answers
31 views

How can I solve for X in a sum function?

I have this inequality constrain, let's call it (*) $$\frac{1}{C}\sum_{i=1}^{n} X_{i}Y_{i} < Z$$ (forgive me if it should be $i=0$ and $n-1$, this is my first question here and I'm very rusty) ...
2
votes
5answers
95 views

How to sum:$\sum_{k=1}^{n}(k²-k)$ without using this: $\sum_{k=1}^{n}(k²)=\frac{n(n+1)(2n+1)}{6}$?

I have tried to find the sum of this series $\sum_{k=1}^{n}(k^2-k)$ without using $\sum_{k=1}^{n}(k^2)=\frac{n(n+1)(2n+1)}{6}$ but i can't . Can anyone explain this to me if there is a way to do ...
2
votes
2answers
48 views

Simplify sum $\sum_{i=0}^k(-1)^ii\binom{n}{i}\binom{n}{k-i}$ for $n\geq k\geq 0$

The problem asks us to simplify the following sum: $$\sum_{i=0}^k(-1)^ii\binom{n}{i}\binom{n}{k-i}$$ for $n\geq k\geq 0$. I've tried the following: ...
7
votes
1answer
52 views

Partitioning $n$ naturals summing $2N$ into two sets summing $N$

I'm trying to solve this problem: Let $a_1, \ldots , a_n$ be natural numbers such that $a_k \le k$ for every $k = 1,\ldots,n$, and $\sum_{k=1}^{n} a_k=2N$. Show that there exists a partition of ...
3
votes
1answer
49 views

Find this closed form $\sum_{k=1}^{n}\left(\lfloor a_{k}\rfloor +\lfloor a_{k}+\frac{1}{2}\rfloor \right)$

Let $\dfrac{1}{a_{k}}=\dfrac{1}{k^2}+\dfrac{1}{k^2+1}+\cdots+\dfrac{1}{(k+1)^2-1}$ I need some ideas to exploit for finding the closed form of $$\sum_{k=1}^{n}\left(\lfloor a_{k}\rfloor +\lfloor ...
2
votes
0answers
36 views

Summation Formula Interpretation involving Roots of Unity

In a paper on Lacunary Recurrence Relations, D. H. Lehmer is beginning his preliminary information and presents the following sum formula: Let $p,q,r,s$ be positive integers, and let $n,t$ be ...
2
votes
1answer
70 views

Sum with binomial coefficients and integer powers

I would like to have an analytic expression for the following sum $$ G_{n,a} = \sum_{p=1}^n \frac{(-1)^p p^{2(a+n)}}{(n-p)! (n+p)!} \;. $$ I am not sure it has a closed form, but I would at least ...
1
vote
1answer
19 views

if $Y\sim \textrm{Bin}(n,\alpha) \space,\space X\mid Y\sim \textrm{Bin}(Y,\beta)$ then $X\sim \textrm{Bin}(n,\alpha \beta)$

Suppose \begin{equation*}Y\sim\textrm{Bin}(n,\alpha)\space,\space (X\mid Y)\sim\textrm{Bin}(Y,\beta)\end{equation*}I'm trying to prove that $X\sim\textrm{Bin}(n,\alpha \beta)$. I began with saying ...
3
votes
3answers
51 views

Computing sum with $\cot$

I want to compute terrible sum: $$ z\cot z = \frac{z}{2^n}\left(\cot\frac{z}{2^n}-\tan\frac{z}{2^n} + \sum_{k=1}^{2^{n-1}-1} \cot\left(\frac{z + k\pi}{2^n}\right) + \cot\left(\frac{z - ...
7
votes
3answers
63 views

Closed form for the sum $\sum_{k=0}^n a^k \left\lfloor\frac{k}{p}\right\rfloor$

I want a closed form for the sum $$S=\sum_{k=0}^n a^k \left\lfloor\frac{k}{p}\right\rfloor$$ where: $a\ne 1<p<n;\quad p\in\mathbb Z$ I know a related identity, $$\quad\displaystyle ...
5
votes
4answers
48 views

How to determine the number removed from the list [duplicate]

One number is removed from a set of integers from 1 to n,the average of the remaining numbers is $\large{\frac{163}{4}}$. Which number was removed? I tried to find the mean of ...
0
votes
2answers
81 views

Prove by induction that $\sum_{i=1}^{n} 2i=(n+1)n$, for every positive integer n. [duplicate]

Can anyone explain the concept behind this? I just don't get how I should proceed with it? Like each step, why and how is it done? Prove by induction that $\displaystyle\sum_{i=1}^{n} 2i=(n+1)n$, ...
2
votes
1answer
36 views

Split number into minimum sum components

I was wondering if there is an analytical solution for the following optimization problem? We have a given real number say $k$. It is needed to split $k$ into minimum number of real components, so ...
0
votes
2answers
47 views

Concrete Mathematics Multiple Sum index change

In Concrete Mathematics, after the "Rocky Road" equality, the authors introduce the problem $$\sum_{1\leq j< k\leq n} \frac{1}{k - j}$$ They eventually arrive at the solution involving letting $k ...
1
vote
1answer
31 views

Name and shorthand for finite sum

If we have a name and notation for the product of natural numbers up to and including n (factorial and !), then what is the name and shorthand for the sum of natural numbers up to and including n? (I ...
0
votes
5answers
61 views

Sum of $n$ terms of series

What is the sum of $n$ terms of the series $${1\over(4\times 9)} + {1\over(9\times 14)} + {1\over(14\times 19)} + {1\over(19 \times 24)} + ... ?$$ The answer is $n\over 4(5n+4)$, but I can't figure ...
0
votes
0answers
21 views

Derivation of least squares for a line $y=a+bx$

I was trying to obtain the formula for the least squares regression for a line: I'm not able to compute the formula that gives the errors on the two parameter. For the "true value" I obtained for ...
0
votes
0answers
15 views

Summation closed form for $k^{n^p}$

Is there any closed form expression for $$\large\sum_{n = 0}^{\infty} k^{n^p} / \sum_{n=0}^{\infty} k^{-n^p}$$ I know that for $p=1$, this converges to $-k$. Is there any other known solution for ...
3
votes
1answer
29 views

Sum of the first $n$ terms of polynome

There are several related questions here, but none seem to have helped me so far. There is an exercise in my book that goes like this: Determine the sum of the first $n$ elements of $$x^3 + x^5 + x^7 ...
2
votes
2answers
55 views

Find sum with binomial coefficients and powers of 2

Find this sum for positive $n$ and $m$: $$S(n, m) = \sum_{i=0}^n \frac{1}{2^{m+i+1}}\binom{m+i}{i} + \sum_{i=0}^m \frac{1}{2^{n+i+1}}\binom{n+i}{i}.$$ Obviosly, $S(n,m)=S(m,n)$. Therefore I've tried ...
-2
votes
1answer
31 views

$n k^n$ summation question [closed]

How does one prove that Can this be extended to higher powers such as: Thanks!
1
vote
1answer
48 views

The logic behind rearranging summations - generating functions

I'm learning about generating functions, and one part that I am struggling with is the logic behind rearranging summations (particularly double summations). I'll illustrate an example: Using the ...
1
vote
4answers
253 views

How to simplify this triple summation containing binomial coefficients?

$$ \large\sum_{i=0}^{n} \sum_{j=i}^{n} \sum_{k=j}^{n} \binom{i+m-1}{m-1}\binom{j+m-1}{m-1}\binom{k+m-1}{m-1} $$ How to solve it when this involve more than thousand summation ?
0
votes
1answer
27 views

Ways to further simply recursive relation

I was working on a power series solution in my ODE class and I had found that my $a_n$ seemed to be defined as $$a_n=\frac{a_o}{(n^{2}(n-1)^{2}…(n-(n+2))^{2}}$$ but I am having trouble understanding ...
3
votes
2answers
120 views

Is there a closed form for $\sum_{n=0}^{+\infty} \frac{1}{\sqrt {n!}}$?

This is a curiosity question. Let's consider the following sum: $$S=\sum_{n=0}^{+\infty} \frac{1}{\sqrt{ n !}}$$ The question asked to prove its convergence, which I did using the ratio test. So I ...
4
votes
0answers
117 views

How to solve this multiple summation?

How to solve this summation ? $$\sum_{0\le x_1\le x_2...\le x_n \le n}^{}\binom{k+x_1-1}{x_1}\binom{k+x_2-1}{x_2}...\binom{k+x_n-1}{x_n}$$ where $k$ , $n$ are known. Due to hockey-stick identity , ...
2
votes
1answer
25 views

Finding a bound on double summation involving primes

I am reading a number theory proof of a result in which I am stuck on a bound.Suppose $p_1$ and $p_2$ are primes with the property that each $p_i$ satisfies $e^r \leq P_i <e^{r+1}$ and $P_1 \equiv ...
2
votes
1answer
60 views

Evaluate $\lim\limits_{x\to\ 1}\frac{x+x^2+\cdots+x^n-n}{x-1}$ without L'Hospital's rule [duplicate]

I used substitution $$t=x-1, x=t+1,x\rightarrow1\Rightarrow t\rightarrow 0$$ Now the expression is $$\lim_{t\to\ 0}\frac{t+1+(t+1)^2+\cdots+(t+1)^n-n}{t}$$ Can we use the sum of geometric sequence ...
1
vote
2answers
47 views

Proof by counting two ways

Proof by counting two ways: \begin{equation}\sum_{k_1+k_2+...+k_m=n}{k_1\choose a_1}{k_2\choose a_2}...{k_m\choose a_m}={n+m-1\choose a_1+a_2+...+a_m+m-1}\end{equation} I have a proof by algebra for ...
2
votes
4answers
86 views

Find $x_{n}$ if $x_{1}=a>0$ and $x_{n+1}=\frac{x_{1}+2x_{2}+…+nx_{n}}{n}$

Find $x_{n}$ if $x_{1}=a>0$ and $x_{n+1}=\frac{x_{1}+2x_{2}+...+nx_{n}}{n}$ I have a problem finding sum of $$x_{1}+2x_{2}+...+nx_{n}$$ I don't see the term $x_{2}$ because if $x_{1}=a$ for ...
3
votes
1answer
36 views

“Commutativity” of sums

In general, is $\sum_i \sum_j f(i,j) = \sum_j\sum_i f(i,j)$ ? With $f(i,j)$ I mean some expression that depends on $i$ and $j$. If yes, how could I prove that?
8
votes
2answers
89 views

Rough bound for sum $\binom{3n}{0}+\binom{3n}{1}+\cdots+\binom{3n}{n-1}$

Is it true that $$\frac{\dbinom{3n}{0}+\dbinom{3n}{1}+\cdots+\dbinom{3n}{n-1}}{2^{3n}}<\frac13$$ for all positive integers $n$? I've plotted the first few values of $n$ and noticed that the ...
2
votes
3answers
31 views

$\sum^6_{i=1}(x_i-\bar{x})^2$ as $\sum^6_{i=1}x_i^2 - 6\bar{x}^2$ what rules where applied?

consider the set $X = \{20, 30, 40, 50, 60, 70\}$ and the mean $\bar{x} = 45$ then $\sum^6_{i=1}(x_i-\bar{x})^2 = 1750 = \sum^6_{i=1}x_i^2 - 6\bar{x}^2$. How would I transform the first term by hand ...
1
vote
2answers
47 views

Ratio of two summations

I devised this question based on recent (and not-so-recent) MSE questions on summations. Evaluate ...
0
votes
1answer
23 views

time complexity of an algorithm

Hi all i'm trying to predict/calculate the time complexity of an algorithm but i'm having some difficulties with the summations the algorithm: ...
2
votes
0answers
21 views

evaluation summation Erlang distribution

I have to calculate a renewal function: $H(t)$. In which: $$ H(t)=\sum_{n=1}^\infty \left(1-e^{-\lambda t}\sum_{i=0}^{2n-1}\frac{(\lambda t)^i}{i!}\right) $$ I think it can be solved by switching ...
1
vote
2answers
35 views

Which is greater ? Sum of odd power terms or even power terms in the exponential Taylor series?

I came across this question, in a book. Define $f(x) = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{{(2n+1)}!} $ and $ g(x) = \sum_{n=0}^{\infty} \frac{x^{2n}}{{(2n)}!} $, where x is a real number. Then, ...