All Questions
Tagged with summation algebra-precalculus
977
questions
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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 ...
1
vote
2
answers
58
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How to simplify double summation in a sophisticated way?
How can I simplify the double summation
$$\sum^{k-1}_{i=1}\sum^i_{j=1}\left( e ^{-\beta(t_{i+1}-t_j)}-e ^{-\beta(t_i-t_j)}\right)$$
to have only one summation in a sophisticated way? I know a lot of ...
- 619
3
votes
0
answers
100
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Simplifying a subindex equation
Consider the following equation
$$
\frac{e^{\sum_i \alpha_i x_i}}{\sum_i x_i}=\sum_k\frac{e^{\sum_i \alpha_i y_{i, k}}}{\sum_i y_{i, k}}
$$
Is it possible to write $x_i$ as a function of the terms $y_{...
- 3,435
2
votes
0
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60
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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 ...
- 3,435
0
votes
1
answer
48
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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^{\...
- 3,435
2
votes
2
answers
97
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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
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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
398
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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
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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
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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
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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
$...
- 5,450
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votes
1
answer
77
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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
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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)+...
- 7,031
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votes
1
answer
59
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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
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$y_i \left[ 1 - \log \frac{t_i}{y_i} - \sum_{j=1}^C y_j \log \frac{t_j}{y_j}\right] = t_i \iff y_j=t_j \ \forall \ j=1,2,...,C$
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
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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
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0
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139
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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 $2$ ...
1
vote
1
answer
41
views
Iverson bracket - infinite additivity for pairwise disjoint sets
Suppose we have a sequence of mutually (pairwise) disjoint sets/events $B_1, B_2, B_3, ... $
EDIT: The sets $B_i$ ($i=1,2,3,\dots$) are subsets of some set $\Omega$.
For the Iverson bracket, is the ...
- 12.6k
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2
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92
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Can someone give me a hint to this question concerning $\sum_{i=1}^n |x-i|$?
Find the smallest positive integer $n$ for which
$|x − 1| + |x − 2| + |x − 3| + · · · + |x − n| \geq 2022$
for all real numbers $x$.
I don't think I can combine any of these terms, right? So I started ...
- 13
3
votes
3
answers
168
views
find a closed form formula for $\sum_{k=1}^n \frac{1}{x_{2k}^2 - x_{2k-1}^2}$
Let $\{x\} = x-\lfloor x\rfloor$ be the fractional part of $x$. Order the (real) solutions to $\sqrt{\lfloor x\rfloor \lfloor x^3\rfloor} + \sqrt{\{x\}\{x^3\}} = x^2$ with $x\ge 1$ from smallest to ...
- 2,031
3
votes
2
answers
145
views
Sum of reciprocal of primes failed computation
Set $$X:=\sum_p \dfrac{1}{p^2}=\dfrac{1}{2^2}+\dfrac{1}{3^2} +\dfrac{1}{5^2}+\cdots$$
As $X$ is absolute convergent and less than $1$, we have (not sure for infinite rearrangement) naive calculation ...
- 1,033
-3
votes
1
answer
79
views
Evaluating $\sum_{j=i+1}^n 1$ [closed]
We know that:
$$\sum_{j=0}^n j$$
we can evaluate it with the formula:
$$\frac{n(n+1)}{2}$$
so how would we evaluate this sum?
$$\sum_{j=i+1}^n 1$$
yes it is a $1$ and not $J$.
- 35
0
votes
1
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38
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How to simplify: $C_{t} = r[a_{t} + \frac{1}{1+r} * \sum_{j=0}^{\infty} (\frac{1}{1+r})^j * E(w_{t+j})]$
I have to find C_t (Optimal Consumption for each period). Thank you!
$$C_{t} = r[a_{t} + \frac{1}{1+r} * \sum_{j=0}^{\infty} (\frac{1}{1+r})^j * E(w_{t+j})]$$
Where,
$$w_{t+j} = \begin{cases} w + \...
-1
votes
1
answer
75
views
how to find the sum of these terms without the gamma function?
While solving a problem based on integration, I arrived at the following
$$\sum\limits_{x = 1}^{38} \ln\left(\frac{x}{x+1}\right)$$
I'm supposed to prove that this is less than $\ln(99)$
in order to ...
- 1,346
1
vote
1
answer
74
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Simplifying $\sum\limits^n_{i=1}\bigl(\prod\limits^n_{j=1}\bigl(a_j\bigl\lfloor\frac{x_j-x_i}{|x_j-x_i|+1} \bigr\rfloor+1\bigr)\bigr)b_i$
I have an indexed finite set of elements $X = \{x_1,x_2,x_3,...,x_n\}$, where $x_i \in \mathbb{R}$. And a corresponding indexed finite set $A = \{a_1,a_2,a_3,...,a_n\}$, where $a_i \in [0,1]$ and a ...
- 197
-5
votes
2
answers
100
views
Calculate $\frac {1}{2\cdot3\cdot 4}+\frac {1}{3\cdot4\cdot 5}+\frac {1}{4\cdot5\cdot 6}+\frac {1}{5\cdot6\cdot 7}$ [closed]
Calculate the following sum.
$$\frac {1}{2\cdot3\cdot 4}+\frac {1}{3\cdot4\cdot 5}+\frac {1}{4\cdot5\cdot 6}+\frac {1}{5\cdot6\cdot 7}$$
My attempt
$$\sum =\frac {5\cdot 6\cdot 7+2\cdot6\cdot 7+2\...
- 652
7
votes
1
answer
184
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Show that $|x_{k+1}-x_k| \leq 1$ (for $0<k<n$) implies $\sum_{k=1}^n |x_k| - \left|\sum_{k=1}^n x_k\right|\leq\lceil(n^2-1)/4\rceil$.
Let $n\ge 1$ be a positive integer and let $x_1,\cdots, x_n$ be real numbers so that $|x_{k+1}-x_k|\leq 1$ for $k=1,2,\cdots, n-1$. Show $$\sum_{k=1}^n |x_k| - \left|\sum_{k=1}^n x_k\right|\leq \left\...
- 1,837
2
votes
1
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88
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Best way to solve a summation with binomial coefficients in denominator apart from telecoping method
The value of $\sum_{r=1}^{m}\frac{(m+1)(r-1)m^{r-1}}{r\binom{m}{r}} = \lambda$ then the correct statement is/are
(1) If $m=15$ and $\lambda$ is divided by m then the remainder is 14.
(2) If $m=7$ and $...
- 445
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votes
1
answer
55
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How can $\frac1s(\sum_{k=0}^\infty \frac{s^k}{k!}-1) = \frac1s(\sum_{k=1}^\infty \frac{s^k}{k!})$?
I'm having trouble with a proof of the moment generating function via Taylor series in Introduction to probability, statistics and random processes which, in relevant part, states the following:
$$\...
- 199