All Questions
Tagged with summation algebra-precalculus
974
questions
2
votes
0
answers
60
views
How to simplify $e^{-\sum_{i=-k}^k(k-|i|)x_i}$?
Consider the expression given by
$$
\large e^{-\Large\sum_{i=-k}^k(k-|i|)x_i}
$$
Is there a way of simplifying this expression?
For example, provided $\{x_i\}$ is bounded and "smooth" enough ...
- 3,405
0
votes
1
answer
48
views
How to simplify $\sum_{k=0}^{n} \frac{e^{\sum_{i=0}^kx_{i}}}{\sum_{i=0}^k x_{i}}$
Is it possible to simplify
$$
S(\mathbf{x})=\sum_{k=0}^{n} \frac{e^{\sum_{i=0}^kx_{i}}}{\sum_{i=0}^k x_{i}}
$$
A few observations:
$\sum_{k=0}^{n}\sum_{i=0}^k x_{i}=\sum_{k=0}^{n}(n+1-k)x_k$
$ e^{\...
- 3,405
2
votes
2
answers
96
views
what formula should I use to found the sum of $Σ_{ k = 2 }^{ 6 } ( { k }^{ 2 } -7) $
Today I am trying to solve this question, and it asks me to found the sum.
$$Σ_{ k = 2 }^{ 6 } ( { k }^{ 2 } -7) $$
I tried to found a formula to solve this, but I could not found it. In this ...
- 143
3
votes
1
answer
82
views
Does $\sum_{j=1,j\neq k}^n\frac{z_k}{z_k-z_j}=\sum_{j=1,j\neq m}^n\frac{z_m}{z_m-z_j}$ implies the $(z_j)_{j=1,...,n}$ are the nth roots of $z_1^n$?
Let $(z_1,\ldots z_n)\subseteq \mathbb{C}.$
Does
$$
\sum_{j=1\,,\,j\neq k}^n \frac{z_k}{z_k-z_j}=\sum_{j=1\,,\,j\neq k'}^n \frac{z_{k'}}{z_{k'}-z_j}
$$ for all $k,k'\in\{1,...,n\}\,$, implies that ...
- 87
0
votes
1
answer
391
views
Find a simple formula for $\sum_{k=1}^{n}(5k + 1)$
$$\sum_{k=1}^{n}(5k + 1) = \sum_{k=1}^{n}5k + \sum_{k=1}^{n}(1) = \sum_{k=1}^{n}5k + n = 5\frac{n(n+1)}{2} + n = 5n^2/2 + 5n/2 + n$$
Can something further be done ?
- 1,135
1
vote
2
answers
120
views
How could I solve $\sum_{k=3}^7 k^2-1$ by using the formula
This afternoon I am dealin with with this question:
$$\sum_{k=3}^7 k^2-1$$
When I using Microsoft Math solver for this question or just plug in every numbers, I get 130. However, when I using the ...
- 143
0
votes
0
answers
102
views
Is there a formula for $\sum_{i=1}^n \frac 1i$? [duplicate]
Is there a formula for
$$\sum _{i=1}^n\frac{1}{i} \,?$$
Any help would be great
- 17
3
votes
2
answers
94
views
The sum of the numbers from $100$ to $999$ that do not have the digit $0$ as well as do not have repeated digit.
Considering the numbers from $100$ to $999$. Excluding numbers that have the digit $0$, also excluding numbers that have repeated digit. What is the sum of remaining numbers?
That is, we need to find
$...
- 5,468
0
votes
1
answer
76
views
find the smallest possible value of m so that there are real numbers $b_j$ satisfying $f_9(n) = \sum_{j=1}^m b_j f_9(n-j)$ for $n>m$
For an integer n, let $f_9(n)$ denote the number of positive integers $d\leq 9$ dividing n. Suppose $m$ is a positive integer and $b_1,b_2,\cdots, b_m$ are real numbers so that $f_9(n) = \sum_{j=1}^m ...
- 2,031
-1
votes
2
answers
70
views
Determine $n$ in the equality $1+2(2!)+3(3!)+…+n(n!)=719$ [duplicate]
Determine $n$ in equality: $1+2(2!)+3(3!)+…+n(n!)=719$.(Answer:5)
I tried to separate the terms but I didn't succeed
$ 1+2(2!)+3.3(2!)+4.4.3(2!)+\ldots+n.n((n-1)!)=719\\ 2!(2+3^2+4^2.3+5^2.4.3\ldots)+...
- 6,849
0
votes
1
answer
59
views
Can you help me solve this summation?
I've added an image of how I've approached this problem.
Any clarity would be appreciated.
- 73
1
vote
2
answers
58
views
$y_i \left[ 1 - \log \frac{t_i}{y_i} - \sum_{j=1}^C y_j \log \frac{t_j}{y_j}\right] = t_i \iff y_j=t_j \ \forall \ j=1,2,...,C$
I'm trying to prove the following affirmation:
Given the constants $t_j \in (0,1)$ and the variables $y_j \in(0,1)$, for $j=1,2,...,C$:
\begin{equation}
y_i \left[ 1 - \log \frac{t_i}{y_i} - \sum_{...
- 13
1
vote
0
answers
39
views
Is this a valid proof method (and how to complete it) of the exchange of order of indexed summations?
I want to prove that
$$
\sum_{1 \leqq i \leqq a} \sum_{1 \leqq j \leqq b} f(i, j) = \sum_{1 \leqq j \leqq b}\sum_{1 \leqq i \leqq a} f(i, j)
$$
Which is a very simple statement, but also a bit vexing ...
- 193
2
votes
0
answers
39
views
Simplifying $\sum_{k=1}^{\log_{2}{n}}\log_{3}{\frac{n}{2^k}}$ in two ways gives different results
I want to calculate the result of
$$\sum\limits_{k=1}^{\log_{2}{n}}\log_{3}{\frac{n}{2^k}}$$ I used two below approaches. Both approaches are based on $\log A + \log B = \log (A \times B)$ and $\sum\...
- 485
1
vote
0
answers
138
views
Solving a geometric-harmonic series
Find the value of $\displaystyle \frac21- \frac{2^3}{3^2}+ \frac{2^5}{5^2}- \frac{2^7}{7^2}+ \cdots$ till infinite terms.
found this problem while integrating $\arctan\left(x\right)/x$ from $0$ to $2$ ...
