All Questions
Tagged with summation algebra-precalculus
977
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Finding the formula for the summation $\sum_{i=1}^n (2i-1) = 1+3+5+...+(2n-1)$.
I'm going through Calculus by Spivak and one of the questions is to find a formula for a summation.
$$\sum_{i=1}^n (2i-1) = 1+3+5+...+(2n-1).$$
I got the correct answer, $n^2$, but did it an ...
CommunityBot
- 1
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1
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Formula to increase x by y z times [closed]
What is the formula to increase x by y z times?
For example the number 4 I want to increase by 4 50 times. I added it out (4+8+12+16+20+24 etc....to 204) and got the right answer 5304 but what is ...
- 3
2
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4
answers
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How did Rudin change the order of the double sum $\sum_{n=0}^\infty c_n\sum_{m=0}^n\binom nma^{n-m}(x-a)^m$?
I see many people change the order of sum but I don't understand how they did that.
Is there is a way to change the order of the sum, $\sum\limits_{k=a}^n\sum\limits_{j=b}^m X_{j,k}$ and $\sum\...
- 1,637
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3
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Show $\sum_{i=0}^n{i\frac{{n \choose i}i!n(2n-1-i)!}{(2n)!}}=\frac{n}{n+1}$
How can this identity be proved?
$$\sum_{i=0}^n{i\frac{{n \choose i}i!n(2n-1-i)!}{(2n)!}}=\frac{n}{n+1}$$
I encountered this summation in a probability problem, which I was able to solve using ...
- 12k
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1
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Proof that the square root of the mean of the squares is always greater than or equal to the mean of weighted values
I couldn’t think of a better title, but basically you are given some values $x_1$, $\ldots$, $x_n$ and some weights $p_1$, $\ldots$, $p_n$ (with $x_k\in\mathbb{R}$ and $p_k\in[0,1]$, also $p_1+\ldots+...
- 4,011
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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
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Simple algebra in rearring terms
I have a very simple mathematical question, and it is just about algebra which seems very tedious. First, let me state my problem from the beginning:
Let $i$ be an index representing countries ($i = {...
- 105
1
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3
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I want to use integration for performing summation in Algebra
I am a class 9th student. Sorry if my problem is silly.
I am trying to find the sum of squares from 1 to 10. For this I tried summation, and it was fine.
But now I came to know that Integration can be ...
- 268k
2
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1
answer
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Let $A_{k}=\{0,... ,n\}\setminus\{k\}.$ How to prove $\sum_{k=0}^{n}\left[(-1)^{k+1}\prod_{\substack{i,j\in A_{k}\\i<j}}(a_{i}-a_{j})\right]=0$?
Let $A_{k}=\{0,1,\ldots,n\}\setminus\{k\}$ for each $k=0,1,\ldots ,n$.
I think that the following equality is true for all $n\in\mathbb{N}, n\geq 2$ :
\begin{align}
\sum_{k=0}^{n}\left[(-1)^{k+1}\...
- 23
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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
3
votes
1
answer
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Is there a sequence where the average of the squares is proportional to the square of the average?
Usually in math, the order of operations matters when you do anything more complicated than addition or multiplication. If you have a list of numbers, whether you apply the average or the squaring ...
- 93.3k
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Restructuring Jacobi-Anger Expansion
In Jacobi-Anger expansion, $$e^{\iota z \sin(\theta)}$$ can be written as:
$$e^{\iota z \sin(\theta)} = \sum_{n=-\infty}^{\infty} J_n(z)e^{\iota n \theta}$$
where $J_n(z)$ is the Bessel function of ...
- 31
2
votes
1
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Double Sum to Product Derivation
The function after the double-sigma sign can be separated into the
product of two terms, the first of which does not depend on $s$ and
the second of which does not depend on $r$. Source
Is the ...
- 71
6
votes
4
answers
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if 1, $\alpha_1$, $\alpha_2$, $\alpha_3$, $\ldots$, $\alpha_{n-1}$ are nth roots of unity then...
if 1, $\alpha_1$, $\alpha_2$, $\alpha_3$, $\ldots$, $\alpha_{n-1}$ are nth roots of unity then
$$\frac{1}{1-\alpha_1} + \frac{1}{1-\alpha_2} + \frac{1}{1-\alpha_3}+\ldots+\frac{1}{1-\alpha_n} = ?$$
...
- 3,310
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Proof that $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$
Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ?
- 61.5k