All Questions
Tagged with summation algebra-precalculus
977
questions
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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
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0
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39
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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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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
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1
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41
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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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93
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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
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168
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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
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2
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145
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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
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79
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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
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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,
$$w_{t+j} = \begin{cases} w + \...
-1
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1
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75
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how to find the sum of these terms without the gamma function?
While solving a problem based on integration, I arrived at the following
$$\sum\limits_{x = 1}^{38} \ln\left(\frac{x}{x+1}\right)$$
I'm supposed to prove that this is less than $\ln(99)$
in order to ...
- 1,346
1
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1
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74
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Simplifying $\sum\limits^n_{i=1}\bigl(\prod\limits^n_{j=1}\bigl(a_j\bigl\lfloor\frac{x_j-x_i}{|x_j-x_i|+1} \bigr\rfloor+1\bigr)\bigr)b_i$
I have an indexed finite set of elements $X = \{x_1,x_2,x_3,...,x_n\}$, where $x_i \in \mathbb{R}$. And a corresponding indexed finite set $A = \{a_1,a_2,a_3,...,a_n\}$, where $a_i \in [0,1]$ and a ...
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2
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100
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Calculate $\frac {1}{2\cdot3\cdot 4}+\frac {1}{3\cdot4\cdot 5}+\frac {1}{4\cdot5\cdot 6}+\frac {1}{5\cdot6\cdot 7}$ [closed]
Calculate the following sum.
$$\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
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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 $$\sum_{k=1}^n |x_k| - \left|\sum_{k=1}^n x_k\right|\leq \left\...
- 1,837
2
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1
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88
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Best way to solve a summation with binomial coefficients in denominator apart from telecoping method
The value of $\sum_{r=1}^{m}\frac{(m+1)(r-1)m^{r-1}}{r\binom{m}{r}} = \lambda$ then the correct statement is/are
(1) If $m=15$ and $\lambda$ is divided by m then the remainder is 14.
(2) If $m=7$ and $...
- 445
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1
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How can $\frac1s(\sum_{k=0}^\infty \frac{s^k}{k!}-1) = \frac1s(\sum_{k=1}^\infty \frac{s^k}{k!})$?
I'm having trouble with a proof of the moment generating function via Taylor series in Introduction to probability, statistics and random processes which, in relevant part, states the following:
$$\...
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