All Questions
23
questions
3
votes
4
answers
160
views
Proving $\sum_{i=0}^n (-1)^i\binom{n}{i}\binom{m+i}{m}=(-1)^n\binom{m}{m-n}$
I am trying to prove the following binomial identity:
$$\sum_{i=0}^n (-1)^i\binom{n}{i}\binom{m+i}{m}=(-1)^n\binom{m}{m-n}$$
My idea was to use the identity
$$\binom{m}{m-n}=\binom{m}{n}=\sum_{i=0}^n(-...
- 87
0
votes
0
answers
104
views
series based on $(1+x+x^2)^n$
Question : Let $a_r$ denote the following $$(1+x+x^2)^n=\sum_{r=0}^{2n}a_rx^r$$
then prove the following
$$\sum_{r=0}^{n}(-1)^r\binom n r a_r = \begin{cases}
0 & n \ne 3k \text{ for all ...
- 931
3
votes
1
answer
139
views
Double summation with binomial coefficients
Question :
Find the value of the following expression :
$$ \frac{\sum_{i=0}^{2024}\sum_{r=0}^{2024}(-1)^r{2024 \choose r}(2024-r)^i}
{\sum_{r=0}^{2025}(-1)^r\binom{2025}{r}(2025-r)^{2025}} $$
I am not ...
- 37
0
votes
1
answer
51
views
Simplifying Expressions involving Sigma
After reviweing the solutions to a question involving the Biomial Theorem, I arrived at a step, where i was unsure how it occured.
Specifically, i was confused about the logic of:
k=0 -> k=1
n-1 -&...
3
votes
2
answers
117
views
$1\binom{20}1+2\binom{20}2+3\binom{20}3+\dots+19\binom{20}{19}+20\binom{20}{20}$
$$1\binom{20}1+2\binom{20}2+3\binom{20}3+\dots+19\binom{20}{19}+20\binom{20}{20}$$
I solved it by letting the sum be $S$, then adding the sum to itself but taking the terms from last to first and then ...
1
vote
0
answers
48
views
seperating two variables in a function with summation
I'm building a data analysis program that perform on big chunks of data, the issue I'm having is the speed of some operations; to be exact I have a function that takes two variables in this form : $$f(...
- 113
3
votes
2
answers
315
views
Solving binomial summation $\sum_{k=0}^{\lfloor{n/2}\rfloor} \binom{n-k}{k} 2^{n-k}$
How can we solve the sum
$$\sum_{k=0}^{\lfloor{n/2}\rfloor} \binom{n-k}{k} 2^{n-k}$$
The problem arose from a counting question, but I am unable to solve this sum.
Edit:
The counting problem was ...
- 1,334
1
vote
3
answers
73
views
Writing $(m+2)^n-(m-2)^n$ in summation notation.
I have expanded $(m+2)^n-(m-2)^n$ the following way:
$$(m+2)^n-(m-2)^n = 2 {n \choose 1}m^{n-1}+ \dots + {n \choose n-1}m2^{n-1}-{n \choose n-1}m(-2)^{n-1}+{n \choose n}2^n - {n \choose n}(-2)^n$$
...
- 221
2
votes
1
answer
59
views
Roots of a summation
I'm trying to solve an equation:
$$\sum_{n=0}^b \left(\left(\frac{a+xn}{b}\right)\binom{b}{n}(-x)^{b-n}\right)=0$$
Where a and b are constants.
I thought of solving it by using the binomial theorem. ...
- 119
2
votes
2
answers
46
views
How to write $\left(\frac{A+Bs}{C+Ds}\right)^N$ as $\sum_{n=-\infty}^{\infty} s^n P_n$?
We are given this thing
$$J=\left(\frac{A+Bs}{C+Ds}\right)^N$$
Where $A,B,C,$ and $D$ are non-zero constants, $N$ is a positive constant.
We are told to find $P_0$ (the coefficient of the term $s^0$...
- 1,519
2
votes
2
answers
70
views
Question on changing the index of summation
$$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}$$
I believe $j = k-1$ though the book does say that.
This is related to proving the binomial ...
- 21
0
votes
0
answers
33
views
Solve for $x$: $\sum_{r=0}^5 {5\choose r} (-1)^rx^{5-r}3^r = 32$
Solve for $x$: $$\sum_{r=0}^5 {5\choose r} (-1)^rx^{5-r}3^r = 32$$
Looks like binomial theorem. So this would simplify to $(-x+3)^5=32$, and solving gives $x=2$. Is this correct?
- 6,915
6
votes
2
answers
144
views
How to show $\sum\limits_{r=0}^n \frac{1}{r!} \lt\left (1 + \frac{1}{n}\right)^{n+1}$ for all $n \ge 1$?
Using the binomial expansion, it is quite is easy to show that $$\left(1+\frac{1}{n}\right)^n \le \sum_{r=0}^{n} \frac{1}{r!} $$ for all $n\in\mathbb{Z^+}$, with equality holds when $n=1.$ (Can it be ...
1
vote
2
answers
150
views
Solve following summation using Binomial theorem
I apologise in advance for posting another one of these homework assignment-questions; this one is pissing me off.
The question is: solve $\sum_{k=0}^n {{n \choose k}} k 4^k $ using the binomial ...
2
votes
1
answer
111
views
Need help with inductive proof of Binomial Theorem
I'm new to math and trying to learn about the Binomial Theorem, by following this tutorial.
I got stuck trying to read the Induction Proof.
They give an example of using the Sum notation:
$$ (x + y)^...
- 23