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

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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 ...
1
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0answers
23 views

What is the genral form of the laurent exspansion of $\frac{1}{(z-\alpha)^n}$

This is a question from a text book (Saff and Snider, Complex analysis for matemetics science and engineering). Obtain the general formula for the laurent expansion of $$ f_n(z) = ...
6
votes
2answers
106 views

Finding minimum from matrix

Consider following $3\times 3$ matrix. $\begin{pmatrix}3&6&9\\ 2& 4 &8\\ 1 &5& 7 \end{pmatrix}$ I need to find combination of three numbers where each number ...
4
votes
1answer
99 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}\: ...
1
vote
2answers
32 views

Need a more compact formula

This is a part of solution of a programming contest problem $$\sum_{i=0}^{k} {x-i \choose 2} $$ given $x-i \ge 2$ is always true. for $k=1$,$(x-1)^2$ $k=2$, $(x-1)^2+((x-2)*(x-3)/2)$ $k=3$, ...
2
votes
2answers
26 views

Integrate a sum (geometric series) round |z| = 1

This is a question from a text book (Saff and Snider, Complex analysis for mathmatics science and engeneering, page 203). Let $$ f(z) = \sum_{k=0}^\infty (k^3/3^k)z^k $$ Evaluate $$ ...
0
votes
3answers
41 views

Summation Of Binomial & Factorial Series

Looking for an explicit formula for the following: $$ S = \sum _{i=j}^n \frac{\binom{i}{j}}{(i+1)!} $$ any Ideas?
3
votes
2answers
54 views

How to notate a While loop?

I've noticed that Sigma notation is a lot like a For loop in programming. What, if anything, can be used to notate a While loop mathematically? In other words, how to you notate the sum of ...
1
vote
2answers
57 views

Mathematically Expressing the Sum of…

I noticed that $$\large{2x-1 = \frac{x}{2} + \frac{\frac{x}{2}}{2} + \frac{\frac{\frac x2}2}2} + \cdots$$ until the output of one of the steps in the pattern equals $1$. Or in other words $2x-1$ is ...
0
votes
0answers
17 views

Find the signs of elements in a list such that their sum is equal to zero

I have a set $X = \{x_1, x_2, \dots x_N\} \in [0;1]^N$ containing $N$ elements, initially all positive. My goal is to find a vector of signs $S = \{s_1, s_2, \dots s_N\} \in \{-1; 1\}^N$ such that: ...
3
votes
1answer
64 views

A proof involving an infinite sum

I am trying to prove that there exist constants $C_1 > 0$, $C_2>0$ such that$$C_1 \log N \geq\sum_{k=1}^\infty(1 - (1- 1/2^k)^N) \geq C_2\log N$$ where $N\in Z^+$. Could you please give me ...
1
vote
1answer
31 views

Correct usage of sum sign

I want to express in a formula that variables with a certain property shall be added. I think it is best expressed by an example. $A$ is a superset of $B$ and $C$. $B$ can have the properties $X$ ...
0
votes
3answers
48 views

Solving for $x$ in an equation with a summation symbol [closed]

How can find the appropriate value of $x$ in the following infinite sum? $$ \sum x + (x+10) + (x+20) + (x+30) + ..... (x+90) = 530$$
1
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2answers
25 views

Bounding a summation by an integral

In progression of another question, my lecturer states $$\sum_{k=N+1}^\infty \frac{1}{k^2} \le \int^\infty_N \frac{1}{x^2}\, {\rm d}x.$$ However we have not covered bounding summations by integrals ...
0
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5answers
67 views

How can I write $\lim_{n\to\infty} \frac{1}{n^6}{(1+2^5+…+n^5)}$ as an Integral?

How can I write this series as a definite integral? $\lim_{n\to\infty} \frac{1}{n^6}{(1+2^5+...+n^5)}$ We've just covered $\lim_{n\to\infty}\int_{a}^{b}f_n=\int_{a}^{b}\lim_{n\to\infty}f_n$ When ...
2
votes
2answers
61 views

How to calculate the Summation??

Can we get the formula in terms of N and k for this summation series? $$ A=\sum_{t=0}^N\sum_{s=0}^t\sum_{r=0}^sk^rk^{s-r}k^{t-s} $$
0
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0answers
20 views

Can I calculate the upper limit of the definite integral in this equation?

while working on an engineering thesis I came to this equation: $$\sum_{k=1}^N\int_0^Te_k(t)dt=s$$ The $e_k(t)$ is erlangian distribution with shape k, N is an integer (its value can be assumed to any ...
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1answer
20 views

Probability Generating Function Attempt

I am trying to find the PGF for the following distribution: $X_1$ has PMF: $\rho(x) = \frac{-p^x}{x\ln(1-p)},n\in \mathbb N$ Attempt: \begin{align*} \phi_{X}(s) &= E[s^x]\\ ...
1
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1answer
36 views

Combining iterations to come up with a final solution

For a game I'm building, I need to know how many tries it will take for a player to roll several specific numbers (say 1-30 out of 100 possible numbers). The numbers can be repeated and only need to ...
4
votes
2answers
178 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
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1answer
37 views

How to simplify this summation, or express as integral? $\sum_{t=-\infty}^\infty\frac1{\sqrt{(t+ax)^2+4}}$

How to simplify this summation, or express as integral? $$f(x)=\sum_{t=-\infty}^\infty\frac1{\sqrt{(t+ax)^2+4}}$$ $a$ is a constant, say, 24. $$f(x)=\sum_{t=-\infty}^\infty\frac1{\sqrt{(t+24x)^2+4}}$$ ...
6
votes
1answer
295 views

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
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2answers
47 views

How to compute $\sum_{n=1}^N e^{-( n-c)^2}$

I have to compute or at least find good upper and lower bounds on \begin{align*} \sum_{n=1}^N e^{-( n-c)^2/b} \end{align*} and \begin{align*} \sum_{n=1}^N ne^{-( n-c)^2/b} \end{align*} where $c$ ...
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
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0answers
16 views

Voting mechanics clarification

I am reading the following paper: http://www.jstor.org/stable/10.1086/587624 and I am confused about one of the definitions at the start of section III. The auther defines $\bar{U} = ...
0
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3answers
81 views

Find the sum $\sum_{k=1}^n k(k+1)2^k$

On a discrete mathematics past paper, I must find the sum $\sum_{k=1}^n k(k+1)2^k$. Could I have a hint/hints for approaching this problem, please? NB: the preceding problem was as follows. Let ...
0
votes
1answer
62 views

What are the four last numbers in the series $1^1 + 2^2 + 3^3 +\cdots+3458^{3458}$?

What are the four last numbers in $1^1 + 2^2 + 3^3 +\cdots+3458^{3458}$ Hello, I have come across this question, and I have no idea how to solve it. What do you guys think?
1
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4answers
45 views

Double summation counting

I have the following: $$T_n=\sum_{j=1}^n\left(\sum_{k=1}^jk\right)$$ I know that the counting of numbers $1$ to $n$ can be expressed as $$\frac{(n+1)n}{2},$$ which leaves me with ...
5
votes
2answers
100 views

Solve the following summation

$S = \dfrac{n \choose 0}{1} + \dfrac{n \choose 1}{2} + \dfrac{n \choose 2}{3}+\dotsb+\dfrac{n \choose n}{n+1}$
0
votes
1answer
15 views

Show $\sum_{d\mid n}\sum_{e\mid(n/d)}\mu(d)f(e))=\sum_{e\mid n}\sum_{d\mid(n/e)}\mu(d)f(e)$

Need verification as to how $$\sum_{d\mid n}\sum_{e\mid(n/d)}\mu(d)f(e))=\sum_{e\mid n}\sum_{d\mid(n/e)}\mu(d)f(e)$$ I am a little unclear as to how this change works, if it could be clarified for ...
1
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1answer
71 views

How do I use the principle of mathematical induction to prove whether or not $\sum_{k=1}^n (-1)^k = \frac{(-1)^n-1}2$ is a true statement?

For all n elements of Natural Numbers,$\sum_{k=1}^n (-1)^k= \frac{(-1)^n-1}2$. I proved p(1) to be true : $\sum_{k=1}^1 (-1)^k = (-1)^1 =-1$. And $\frac{(-1)^1-1}2 = \frac{(-2)}2 = -1$ So P(1) ...
2
votes
2answers
38 views

Summation with Riemann Zeta Function

So the Riemann zeta function $\zeta(s)$ is commonly defined as $\sum \limits_{n=1}^{\infty} n^{-s}$ Now, suppose that $a_k=\zeta (2k).$ How can I find the value of $$\sum \limits_{k=1}^{\infty ...
1
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1answer
35 views

How do you understand renaming of summation variables?

As a part of a Knuth example, I struggle to understand how you flip the index so easily: $$\sum_{0 < j < k}(k-j) = \sum_{0 < k-j < k} j.$$ Why doesn't Knuth exchange the summand with the ...
0
votes
1answer
12 views

Suppose that number of mistakes on a page is a Poisson RV and independent. From $n$ pages, find the expected number with no mistakes?

A textbook has $n$ pages. The number of mistakes on each page is a Poisson RV with parameter $\lambda$ and is independent of the number of mistakes on all other pages. What is the expected number of ...
1
vote
1answer
36 views

Can $s\sum_{n=0}^{y}(t/s)^{n/y} \ge x$ be solved for $y$?

Is it possible to solve the following equation for y? $s\sum_{n=0}^{y}(t/s)^{n/y} \ge x$ I'm trying to write a slot machine program (for a school assignment I'm making harder than it needs to be for ...
3
votes
1answer
54 views

What is the maximum value of the sum $\sum_{i=1}^L(\bar{x}-x_i)$, in this specific case.

Let $x_i$ be a positive real variable, with $i=1,2,...,K$. We denote by $\bar{x}$ the average value of the values $x_1, x_2,...,x_K$. Let $a=\min_i x_i$ and $b=\max_i x_i$, then $x_i \in [a,b]$. My ...
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1answer
20 views

Sum of order statistics

Is there a general expression for the pdf of the sum of the order statistics? Suppose they are all drawn independently from the same distribution.
4
votes
2answers
239 views

Limit of an expression

$$\lim\limits_{n\to\infty}\frac{1}{e^n\sqrt{n}}\sum\limits_{k=0}^{\infty}\frac{n^k}{k!}|n-k|=\sqrt{2/\pi}$$ Is this limit true? I should show limit is true. It is allowed to use computer programs to ...
4
votes
2answers
79 views

How to show $c_n=\frac11 + \frac12 + \cdots + \frac1n - \ln n$ is a sequence of positive numbers? [duplicate]

For $n \in \mathbb{N}$ let $c_{n}$ be defined by $$c_{n}=\frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{n} - \ln n$$ We have to prove that $c_{n}$ is a decreasing sequence of positive numbers. ...
1
vote
1answer
97 views

Evaluate $\sum \sum 1/n^k $

I wanted to evaluate the sum: $$ \sum_{n \ge 2} \left(\zeta(n) - 1\right) $$ I rewrote this as: $$ \sum_{n\ge 2} \sum_{k\ge 2} \frac{1}{n^k} $$ I tried exploiting the symmetry but that didn't seem ...
0
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0answers
15 views

How can I find condidions when one sum is greater than the other?

I have the following two sums: $$ \sum_{j\neq i} Q_{ij}u_i\int g(z, k) f(z; i, j) \textrm{d}z $$ and $$ \sum_{j\neq i} Q_{ij}u_j\int g(z, k) f(z; i, j) \textrm{d}z, $$ where $k$ is a parameter and the ...
2
votes
3answers
129 views

Number of ways to express a number as the sum of different integers

Given a number $n$, then $P_k(n)$ is the number of ways to express $n$ as the sum of $k$ integers. For example $P_2(6)=7$ $0+6=6$ $1+5=6$ $2+4=6$ $3+3=6$ $4+2=6$ $5+1=6$ $6+0=6$ Now I worked ...
3
votes
5answers
345 views

How to compute series

I have to compute the series $\displaystyle\sum_{n=0}^{\infty}{\frac{3^n(n + \frac{1}{2})}{n!}}$. $$\displaystyle\sum_{n=0}^{\infty}{\frac{3^n(n + \frac{1}{2})}{n!}} = ...
-1
votes
1answer
28 views

Closed form of sum

How to summarize \begin{equation*} f(x,y)=\sum\limits_{n=0}^{\infty}\frac{\exp\left(-\frac{(2n+1)\pi y}{d}\right)}{2n+1}\sin\frac{(2n+1)\pi x}{d}? \end{equation*} I tried to calculate $\partial_y f$ ...
1
vote
3answers
81 views

Closed form of a sum of ratios of integers

I am computing in a program this sum (does it have a "name"): $$\sum_{\alpha=2}^{K} \frac{\alpha-1}{\alpha}$$ is there a way to avoid the sum, term by term, and use a more compact closed form ?
2
votes
0answers
32 views

Coin tossing heads-counting game probabilities

Suppose I have a coin for which the probability of heads is $p$. Now suppose I have a two-player game where the first player may flip this coin $m$ times and the second player may flip this coin $n$ ...
3
votes
0answers
22 views

Summation of a function 2

Let $n$ is a positive integer. $n = p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is the complete prime factorization of $n$. Let me define a function $f(n)$ $f(n) = p_1^{c_1}p_2^{c_2}\cdots p_k^{c_k}$ where ...
1
vote
2answers
27 views

Suppose $X$ and $Y$ are independent Poisson random variables. Find the conditional probability mass function $P(X=k\mid X+Y=m)$ [duplicate]

Suppose $X$ and $Y$ are independent Poisson random variables with parameters $\lambda$ and $\mu$, respectively. Find the conditional probability mass function $P(X=k\mid X+Y=n)$. Don't know how to ...
-1
votes
1answer
18 views

Suppose $X$ and $Y$ are independent Poisson random variables. Find the joint probability mass function $P(X=k, Y=m)$. [closed]

Suppose $X$ and $Y$ are independent Poisson random variables with parameters $\lambda$ and $\mu$, respectively. Find the joint probability mass function $P(X=k, Y=m)$. I know what a Poisson ...