All Questions
Tagged with summation algebra-precalculus
90
questions with no upvoted or accepted answers
2
votes
0
answers
78
views
value of $\frac{\sum_{k=0}^r{n\choose k}{n-2k\choose r-k}}{\sum_{k=r}^n {n\choose k}{2k\choose 2r} {(\frac{3}{4})}^{(n-k)}({\frac{1}{2}})^{2k-2r}}$ .
The question requires us to find the value of $\frac{\sum_{k=0}^r{n\choose k}{n-2k\choose r-k}}{\sum_{k=r}^n
{n\choose k}{2k\choose 2r} {\left(\frac{3}{4}\right)}^{(n-k)}\left({\frac{1}{2}}\right)^{...
- 97
2
votes
0
answers
60
views
Finding a formula for a sum that involves binomial coefficients
Is there a formula for this sum:
$$ \sum_{j=0}^k {n \choose j} {n \choose k-j} (-2)^j \left(-\frac13 \right)^{k-j} ?$$
It reminds me to Vandermonde's identity; but as you can see there is a slight ...
- 897
2
votes
0
answers
143
views
I've come up with two ways to evaluate $\sum_{1 \le j<k\le n} \frac{k}{k-j}$ but only one of them works
I've come up with two ways to solve this double sum but only one of them works:
$$ \sum_{1 \le j<k\le n} \frac{k}{k-j}$$
My first approach is to change $k-j$ into a single $k$. So we have the ...
2
votes
2
answers
82
views
Using trigonometric power formulas to derive an identity for $\cos^3(x)$
I am practicing with manipulating sigma notation and binomial coefficients right now. I am using the formula given here to derive the identity for $\cos^3(x)$
The identity for $\cos^3(x)$ is
$$\cos^3(...
- 3,187
2
votes
0
answers
137
views
Can this summation be done without calculator?
Is it possible to perform the summation ,
$$\sum_{i=1}^{\infty} \frac{1}{i^i}$$
without the use of calculator?
It does converge to a finite value = 1.29129...
Wolfram Alpha link to this
Describe the ...
- 1,663
2
votes
0
answers
115
views
Double Summation Multiplication
There is some simplification, similar to Lagrange's identity, for the multiplication of double summation ?
Double Summation:
$\left( \sum\limits_{\substack{m=1}}^N \sum\limits_{\substack{n=1}}^N a_{...
- 63
2
votes
0
answers
38
views
Upper bound on $\frac{v_{1}}{1-c_{1}x}+\cdots+\frac{v_{n}}{1-c_{n}x}$
Let $v_{1},\ldots,v_{n}$ and $c_{1},\ldots,c_{n}$ be real numbers such that $v_{i}=2\alpha_{i}^{2}$ and $c_{i}=2\alpha_{i}$ for some $\alpha\ge 0$. My question is the following: Can I get, for $x\ge 0$...
- 638
2
votes
0
answers
256
views
Foil in a Summation
I have the following summation, where I find the following result
$ \sum_{i}^n {(a+b_i)(c+d_i)} $
$ \sum_{i}^n {(ac+ad_i + bc_i + b_id_i)} $
However others have told me that I am missing a "N" ...
- 85
2
votes
2
answers
70
views
Question on changing the index of summation
$$b(a+b)^m = \sum_{j=0}^m \binom{m}{j}a^{m-j}b^{j+1}= \sum_{k=1}^m \binom{m}{k-1}a^{m+1-k}b^{k}+b^{m+1}$$
I believe $j = k-1$ though the book does say that.
This is related to proving the binomial ...
- 21
2
votes
0
answers
150
views
Evaluate this product $n \times \frac{n-1}{2} \times \dots \times \frac{n-(2^k-1)}{2^k}$
For $k = \lfloor \log_{2}(n+1) \rfloor - 1$ evaluate
$n \times \frac{n-1}{2} \times\frac{n-3}{4} \times \frac{n-7}{8} \times \dots \times \frac{n-(2^{k}-1)}{2^k}$
So the product goes up to $k$ and I ...
- 69
2
votes
0
answers
105
views
How does one change the top number in a summation?
Sorry I do not know the correct term (I am guessing "upper limit"). Here is what I mean.
$$\sum\limits_{i=1}^{\color{red}{17}}\frac{2i}{i+3}$$
The $17$ is what I am talking about as "the top number". ...
- 5,659
2
votes
0
answers
48
views
Are there efficient ways of computing sums that involve trigonometric functions and q-logarithms (Tsallis q-logarithms)?
I am interested in computing the following sum:
\begin{equation}
\sum\limits_{l=1}^k l^{\beta_1} \cos\left(\omega \log_q(\frac{l}{t_c})\right)
\end{equation}
Here $0 < \omega$, $0 < k < ...
- 11.5k
2
votes
0
answers
730
views
Analytical solution for a variable inside of a summation
I am trying to figure out how to solve the following expression for $x$ and I'm surprised that I don't know what to do.
$$\frac{2n}{x} = \sum_{i=1}^{n} \frac{1}{x-y_{i}}$$
We have that $n$ and $x$ ...
- 5,102
1
vote
0
answers
137
views
Simple algebra in rearring terms
I have a very simple mathematical question, and it is just about algebra which seems very tedious. First, let me state my problem from the beginning:
Let $i$ be an index representing countries ($i = {...
- 105
1
vote
0
answers
103
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Restructuring Jacobi-Anger Expansion
In Jacobi-Anger expansion, $$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