All Questions
30
questions
1
vote
3
answers
93
views
Evaluating $\sum_{\substack{i+j+k=n \\ 0\leq i,j,k\leq n}} 1$
I need to find the sum $$\sum_{\substack{i+j+k=n \\ 0\leq i,j,k\leq n}} 1$$
For $n=1$ we have the admissible values of $(i,j,k)$ as: $(1,0,0),(0,1,0), (0,0,1)$ $$\sum_{\substack{i+j+k=1 \\ 0\leq i,j,...
- 910
0
votes
0
answers
78
views
Double Summation over a Subset of a Cartesian Product
From the "Probability & Statistical Inference, 9th edition" by Hogg, Tannis, Zimmerman, it is stated that one of the properties of the Joint Probability Mass Function of Random Variables ...
- 123
1
vote
2
answers
84
views
How to change index in summation
Pease help me understand how they have changed index of summation from r to n here. If we take $$n = r-s$$ how n is changing from -$\infty $ to $\infty$
- 221
0
votes
1
answer
59
views
Can you help me solve this summation?
I've added an image of how I've approached this problem.
Any clarity would be appreciated.
- 73
0
votes
1
answer
89
views
Is it possible to calculate how many digit number I have to write if I write $n^2$ sequence from 1 to $1000^2$ without using calculator?
From the fact that, for any positive integer n, it will require $1+\lfloor\log_{10}(n)\rfloor$ digits to write. When $\lfloor\ \rfloor$ is floor function.
If I write "1,4,9,16,25,...,1000000"...
- 725
1
vote
2
answers
124
views
Changing summation indices
How does one prove the following result, where $x$ is a three-parameter function defined on $\mathbb Z^3$? $$ \Sigma_{\ell=1}^{P}\Sigma^{\ell-1}_{i=0} x(\ell,i,\ell-i) \quad = \quad \Sigma^{P}_{j=1}\...
- 1,139
1
vote
1
answer
196
views
I wish to solve exactly this formula involving sums and products
I was solving a physics exercise and I encountered this formula:
$$\left< n_l \right>=\left[1+\sum_{k\neq l} \left(e^{bN(l-k)}\frac{\prod_{j\neq l} (1-e^{b(l-j)})}{\prod_{j\neq k} (1-e^{b(k-j)})}...
- 56
0
votes
1
answer
4k
views
discrete mathematics question $2-2\times 7+2\times 7^2− \cdots +2(-7)^n = \frac{1-(-7)^{n+1}}{4}$
This question is from the book Discrete Mathematics and its Applications by Kenneth Rosen page 329. Again for question 8 I face the same probelm I know the solution but I do not understand it.
...
0
votes
2
answers
103
views
Small doubt in derivation of double summation formula
I was reading about double summation of series when the variable are dependent.
My book derived the the formula by creating a matrix to identify a pattern in the series:
Then they derive the formula:
...
- 175
2
votes
1
answer
64
views
Sum of even and odd naturals
I want to prove that the sum of the first even $k$ natural numbers is $k^2+k$ given that the sum of the first odd $k$ naturals is $k^2$. So, \begin{align*} \underbrace{1+3+5}_{k=3}&=k^2=3^2=9 \\ &...
- 2,072
-3
votes
1
answer
50
views
Simplifying expression in summation notation [closed]
Can anyone provide some guidance on how to simplify this expression, if at all possible? Thank you.
$$\sum_{i=1}^{n} {n\choose i} (-1)^{i+1} 2^{n^2+2(n-i)}$$
- 6,099
0
votes
1
answer
146
views
Proving that $\sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}{{n \choose 2k}\cdot 3^{n-2k}}=2^{n-1}(2^n+1) $
Prove that $$\sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}{{n \choose 2k}\cdot 3^{n-2k}}=2^{n-1}(2^n+1)\,.$$
Can somebody help me prove this identity?
- 43
3
votes
2
answers
116
views
How make summation for a series which contains arbitrary elements
I am studding a research paper in winch author presented a analytical model for set traversal and different cases of time complexity.
I am not understanding the one point in the model that is related ...
- 185
1
vote
2
answers
142
views
Double summation problem $\sum_{i=1}^{n-1} \sum_{j=i+1}^n X_i X_j.$
I have to calculate this double summation but I am not sure I am doing it the correct way. Could you please help me with it?
The equation is: $$\sum_{i=1}^{n-1} \sum_{j=i+1}^n X_i X_j. $$
So, for ...
- 21
0
votes
2
answers
52
views
Solving expression with multiple summation notations
In the following, I need to solve the expression in terms of $p_o$. I will appreciate any help in this regard.
$$Y = \lim_{J\to100}\sum^{J}_{j=0}\sum^{j}_{m=0}\frac{\sigma^j}{e^\sigma j!(j-m)!}\lim_{...
- 849