All Questions
Tagged with summation algebra-precalculus
90
questions with no upvoted or accepted answers
1
vote
0
answers
46
views
Trying to find a function that converts a binary sequence to a number
Suppose I have the sequence $\langle x_1, ..., x_n \rangle \in \mathbb{N}^n$ that gets encoded to the binary string
$$
\begin{matrix}
\underbrace{11...1} & 0 & \underbrace{11...1} & 0 &...
- 978
1
vote
1
answer
56
views
Given variable $m$, how do I find zeros of a polynomial in terms of $m$?
This is a summation question about a finite series with sum $m$. I'm trying to write a computer program that takes in a given integer $m$ (which represents the sum of a series) and outputs the number ...
- 11
1
vote
1
answer
116
views
Upper bound of a sum of series
How can I find a tight upper bound for the following expression:
$\sum\limits_{i=1}^{k} a_i \sum\limits_{j = 1}^{i} \frac{1}{b_j} = a_1 \frac{1}{b_1} + a_2 (\frac{1}{b_1} + \frac{1}{b_2}) + \dots + ...
- 41
1
vote
0
answers
35
views
Help with simplification of this expression.
So i have derived this expression and would like to simplify it (i.e find an expression purely in terms of J and v).
$$ J\bigg[1+\sum_{n=1}^{J-1}\prod_{k=1}^{n}\frac{k(1-v)(J-k)}{(k+1)(J-k-1 + vk)}\...
- 61
1
vote
0
answers
160
views
How to isolate and solve for k in a Sigma notation probability mass function equation?
"isolate and solve for k:"
$$P(X = k) = \sum_{k=0}^n {{{K \choose k} {{N-K} \choose {n-k}}}\over {N \choose n}}$$
If the above equation is a function of P, how would the equation be stated as a ...
- 11
1
vote
1
answer
30
views
How can i simplify the following term to get the right side?
$$\sum_{h=1}^{L}\frac{W_h^2S_h^2}{n_h}=\frac{1}{n}\sum_{h=1}^{L}{(W_hS_h)}^2$$
where,
$n_h=\frac{n}{\sum_{h=1}^{L}N_hS_h}N_hS_h$ $\quad\text{and}\quad$ $W_h=\frac{N_h}{N}$ $\quad\text{and}\quad$ $\...
- 1,457
0
votes
0
answers
20
views
Combinatorics/ permutations with summations and conditional permutation or rearrangement inequality
This problem was in the Australian Math Olympiad 2024 and was number 3 on the test. I couldn't see the light at the end of the tunnel when solving this problem but rather I could not find a solution ...
0
votes
0
answers
39
views
Simplification of the ratio between series
I have been trying to solve a problem i posed to myself in the applied sciences, and technically, i did (though it is not of any practical use). But the problem is that the solution is, well, not ...
- 1
0
votes
0
answers
45
views
Inequality with Products and Sums
I need help to find a proof for the following inquality.
Assuming that $ 0 \leq c_i \leq 1 $ and $ 0 \leq d_i \leq 1 $, show that
$$
\prod_{i=1}^N (c_i + d_i - c_i d_i) \geq \prod_{i=1}^N c_i + \prod_{...
- 778
0
votes
0
answers
98
views
If $\sum_{i=1}^n x_i \ge a$, then what can we know about $\sum_{i=1}^n \frac{1}{x_i}$?
Suppose that $$\sum_{i=1}^n x_i \ge a$$
where $a>0$ and $x_i\in (0, b]$ for all $i$. Are there any bounding inequalities we can determine for $$\sum_{i=1}^n \frac{1}{x_i}?$$
I understand that $\...
- 1,615
0
votes
0
answers
20
views
Finding a sufficient condition for dividends to be nonnegative
The Harsanyi dividend is defined as follows:
$d_v (S) = \sum_{R \subseteq S} (-1)^{|S|-|R|} v(R)$
Supermodularity is defined as follows, for $S \subseteq T \subseteq N$:
$v(S \cup \{i\}) - v(S) \leq v(...
- 43
0
votes
0
answers
43
views
Compute a tight upper bound of $\sum_{i=1}^{n-1}\frac{1}{3^i\log{n}- 3i}$?
I am trying to compute a tight upper bound of the sum below.
$\sum_{i=1}^{n-1}n\frac{\frac{1}{3^i}}{\log_3{(n/3^i)}}$
I was able to 'simplify' it up to the expression below.
$n\sum_{i=1}^{n-1}\frac{1}{...
- 185
0
votes
0
answers
104
views
series based on $(1+x+x^2)^n$
Question : Let $a_r$ denote the following $$(1+x+x^2)^n=\sum_{r=0}^{2n}a_rx^r$$
then prove the following
$$\sum_{r=0}^{n}(-1)^r\binom n r a_r = \begin{cases}
0 & n \ne 3k \text{ for all ...
- 931
0
votes
0
answers
78
views
Double Summation over a Subset of a Cartesian Product
From the "Probability & Statistical Inference, 9th edition" by Hogg, Tannis, Zimmerman, it is stated that one of the properties of the Joint Probability Mass Function of Random Variables ...
- 123
0
votes
0
answers
17
views
Rewriting the Sum of indexes i and j across the ordered real numbers x(1), x(1) ... x(n) with a modulus consideration
Question is in image (sorry I don't know how to do the type set)
My attempt is halfway here and I got stuck.
Given LHS
= Sum (i=1 to i=n) for { Sum (j=1 to j=n) [x(i) - x(n)] where [u] denotes ...
