Questions tagged [binomial-theorem]
For questions related to the binomial theorem, which describes the algebraic expansion of powers of binomials.
3
votes
1answer
72 views
How to prove the two identities
Let $m,n$ be positive integers and $N=\{1,\ldots, n\}$. Try to prove the two following identites:
For $m<n$, we have
$$\sum_{A \subseteq N} (-1)^{\left| A\right|} \left(\sum_{j \in A} x_j\right)^m ...
0
votes
0answers
15 views
Induction based on combinations and binomial theorem
I was looking at some questions in a Cambridge text and I reached this question however I am at it for 1 hr and can't seem to get the proof right. Any help ?
1
vote
1answer
39 views
Could someone help clarify the meaning of this pre-calculus question on Binomial Expansion?
I'm taking an online pre-calculus course and I'm currently working on a unit about Binomial expansion.
I've come across an oddly worded (at least to me) question on coefficients:
"What coefficient ...
1
vote
1answer
53 views
A lower bound for $(1-e^x)^n$
I want to find a lower bound for
$$(1-e^x)^n$$
$n$ integer, $x$ real, and $1-e^x\geq 0$. One lower bound is (Bernoulli's inequality)
$$(1-e^x)^n\geq 1-ne^x$$ But I need a tighter lower bound that is ...
0
votes
2answers
36 views
What kind of expansion is this?
In a paper I came across an expansion like this:
$$\cos(m\theta) = C_m^0\cos^m(\theta) - C_m^2\cos^{m-2}(\theta)(1-\cos^2(\theta)) + C_m^4\cos^{m-4}(\theta)(1-\cos^2(\theta))^2 + ...
(-1)^nC_m^{2n}\...
0
votes
0answers
24 views
Generalizing $(^4C_0)^2-(^4C_1)^2+(^4C_2)^2-(^4C_3)^2+(^4C_4)^2=\,^4C_2$
I got this question, and I have a bit of an issue with part (b). I managed to prove the attached result; however, it does not really fit the above result (as it said generalise the result and then ...
-3
votes
0answers
27 views
Finding term that is independent of x in the expansion [closed]
Find the Expansion of
$(2+3x^{-2})(x-2x^{-1})^6$
Can you expain it in detail? Struggling with this and exams are nearing it's urgent!
0
votes
1answer
32 views
Sum of weighted binomial coefficients [duplicate]
I am struggling with computing the following sums:
$$\sum_{k=1}^{n}k\binom{n}{k}=\binom{n}{1}+2\binom{n}{2}+...+n\binom{n}{n}$$
and
$$\sum_{k=0}^{n}\frac{1}{k+1}\binom{n}{k}=\binom{n}{0}+\frac{...
2
votes
3answers
60 views
Writing $(m+2)^n-(m-2)^n$ in summation notation.
I have expanded $(m+2)^n-(m-2)^n$ the following way:
$$(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$$
...
0
votes
1answer
86 views
A Subtle Point in Rudin's Proof on $e$
When Rudin proves the theorem $\lim_{n\rightarrow \infty} (1+\frac{1}{n})^n=e$, he uses the following claim:
Let $t_n= (1+\frac{1}{n})^n=1+1+\frac{1}{2!}(1-\frac{1}{n})+\frac{1}{3!}(1-\frac{1}{n})(...
0
votes
1answer
49 views
Polynomial of $n-1$ degree
Given a polynomial in $\mathbb{C}$, I want to show that a polynomial of the form
$$P_n(z)=\sum^n_{i=0} a_iz^i$$
can be decomposed into something like
$$P_n(z)=(z-z_0)\cdot Q_{n-1}(z),$$
where $Q$ is ...
2
votes
1answer
43 views
Which type of random walk has distribution of “scaled” binomial coefficient?
We know that, for a 1D symmetrical random walk, ${\displaystyle p = 1/2, q = 1/2}$, with equal walk step length, after n steps, its probability distribution will be proportional to binomial ...
0
votes
2answers
40 views
How to prove $n^p>\log n$ for a fixed $p\in (0,1)$ when $n\rightarrow \infty$? [duplicate]
I want to prove:
$$n^p>\log n$$
for a fixed $p\in (0,1)$ when $n\rightarrow \infty$.
I test by some big number and it seems true. But how to prove that?
1
vote
2answers
43 views
Proof of Inequality by Binomial Theorem
I have been stuck in this proof for a while. This should not be a long shot.
Claim If $x_n>0$ for all $n\in N$, then $1+nx_n\leq (1+x_n)^n$.
My strategy is to use Induction Theorem and Binomial ...
1
vote
2answers
64 views
How to solve interest problem without using a crazy binomial expansion?
Problem 9.9
Eric deposits X into a savings account at time 0, which pays interest at
a nominal rate of i, compounded semiannually. Mike deposits 2X into a
different savings account at time 0, ...
4
votes
2answers
59 views
In how many ways a student can get $2m $ Marks
An examination contains four Question papers each paper carrying maximum marks as $m$. Find number of ways a student appearing for all the four papers gets a total of $2m$ Marks.
I used generating ...
-2
votes
1answer
42 views
Having trouble with a Binomial proof by mathematical induction question: $\sum_{j=3}^n \binom{j-1}{2} = \binom{n}{3}$ [closed]
I can't work out how to prove this equation is true by proof of mathematical
Use mathematical induction to prove that, for $n \ge 3$
$$\sum_{j=3}^n \binom{j-1}{2} = \binom{n}{3}$$
Please help, ...
3
votes
5answers
112 views
Show $\left(1-\frac x k\right)^k<\left(1-\frac {x}{ k+1}\right)^{k+1}$
Show that $$\left(1-\frac{x}{k}\right)^k<\left(1-\frac{x}{k+1}\right)^{k+1}$$ for $x>0$, and $k \ge 1$, where $k$ is a whole number.
Is it possible to prove this? I can easily prove ...
4
votes
2answers
56 views
Combinatorial proof of Negative Binomial Identity
For the (usual) Binomial theorem with positive integer exponent, there is a well known nice combinatorial proof.
I am eager to learna similar argument for the proof of negative binomial series: $$(1+...
0
votes
2answers
34 views
Why binomial expansion approximation works?
So I have got the expansion of $$ (4-5x)^.5 = 2 + (5/4)x + (25/64)x^2 $$
I am told to use $ x = 1/10 $ to find an approximation of $ \sqrt2 $. I can do this, giving $ 181/128 $, however the last part ...
5
votes
2answers
343 views
Proof that $\sum\limits _{n=1}^{\infty } n\sum\limits _{j=2}^{\infty }{\frac {(-1)^{j-1}}{j^2} \left( 1-{j}^{-1} \right)^{n-1}} = -\frac12$
How can we prove the following?
$-1+\frac1{12}{\pi }^{2}-\frac12\sum\limits_{n=2}^{\infty }\Gamma \left( n+1 \right) \sum\limits_{k=0}^{n+2}
\,{\frac {\zeta \left( k \right) \left( -
1 \right)...
0
votes
0answers
67 views
References for $ \chi(n)=n\sum\limits_{j=2}^\infty\frac {(-1)^{j-1}}{j^2}\left(1-j^{-1}\right)^{n-1}$ in $\zeta$ expansion?
What is the name of these coefficients related to a series expansion for the Riemann zeta function or any other references? Did the author of this paper derive the series or get it from somewhere else?...
2
votes
1answer
57 views
How to simplify $\sum _{l=1} ^{k} \sum_{j_1 \ge 1} \cdots \sum_{j_l \ge 1, j_1 + \cdots + j_l=k} \binom{k}{j_1 \cdots j_l} \binom{-2}{l} 2^l$
I'm trying to simplify the following equation.
$$\sum _{l=1} ^{k} \sum_{j_1 \ge 1} \cdots \sum_{j_l \ge 1, j_1 + \cdots + j_l=k} \binom{k}{j_1 \cdots j_l} \binom{-2}{l} 2^l$$
Or more specifically, I'...
-4
votes
1answer
46 views
How to get the following expression ? 4 [closed]
Given the following expression:
$$ f=\frac{1}{q (p+\sqrt{q})}\frac{1 + \exp(-2 \sqrt{q})}{1+ \frac{p - \sqrt{q}}{p + \sqrt{q}}\exp(-2\sqrt{q})} $$
I want to derive the following expression:
$$ f = \...
1
vote
1answer
59 views
Coefficient of $x^k$ in ${(1 - e^x)}^{-n}$
What is the coefficient of $x^k$ in ${(1 - e^x)}^{-n}$?
This is what I tried-
Using negative binomial expansion and Taylor series expansion of $e^t$,
$${(1 - e^x)}^{-n} = \sum_{i=0}^{\infty} {n+i-1 ...
1
vote
1answer
40 views
Simplify and compute the MGF of $[1-(1-(1-e^{-ax})^{N_1})(1-(1-e^{-ax})^{N_2})]^{M-1} $
Let $X_1$ and $X_2$ two random variable with CDF
\begin{align}
F_{X_1}(x)&=(1-e^{-ax})^{N_1}\\
F_{X_2}(x)&=(1-e^{-ax})^{N^2}
\end{align}
Let $Z$ random variable with CDF
$$
F_Z(z)=[1-(1-F_{...
1
vote
2answers
41 views
Derivation of Binomial theorem
Let $F(x)$ be the function defined by (1):
$$ 1)F(x)=(1-e^{-ax})^N$$
Using the binomial theorem $F(x)$ can be written as (2):
cc\begin{equation}
2)F(x)=\sum_{n=0}^{N}\binom{N}{n}(-1)^ne^{-axn}
\end{...
0
votes
2answers
77 views
Question wrt Binomial theorem
This question may sound too stupid. But I am quite confused by the Binomial theorem
As per my understanding, let us consider
$(x+y)^4$ without the coefficients.
$(x+y)^4 = x^4 + x^3y + x^2y^2 + xy^...
1
vote
2answers
24 views
Evaluating the variance of the biinomial distribution directly
I know that the easy way to evaluate the mean and variance of the Binomial distribution is by considering it as a sum of Bernoulli distributions.
However, I was wondering just for fun if there is a ...
0
votes
5answers
62 views
Problem using binominal theorem? [closed]
I tried to solve this problem but I couldn't so i'm looking for an help.
Is there any two digit natural number $n$ which fits with following statement?
$$n \mid (4^n - 3^n - 1)$$
The hint is ...
8
votes
4answers
770 views
Remainder on division with $22$
What is the remainder obtained when $14^{16}$ is divided with $22$?
Is there a general method for this, without using number theory? I wish to solve this question using binomial theorem only - maybe ...
0
votes
1answer
32 views
A curve C has an equation of $y = f(x)$, where $ f(x) = \frac{4x}{\sqrt{4+x}+\sqrt{4-x}} $, and $ -4\le x\le4. $
This question is Part $A$ of a binomial theorem question. I am unable to continue as I am stuck here. I really appreciate any help.
A curve $C$ has an equation of $y = f(x)$, where
$$ f(x) = \...
1
vote
1answer
59 views
Multinomial theorem & combinatorial probabilities
A set of $n$ people are given some sweets. There is candy A, B and C and each is given with probability $p_A,p_B$ and $p_C$.
I am trying to find the possible combinations of this system.
We can ...
0
votes
1answer
40 views
Is it possible to extract a variable out of a bounded sum of binomial coefficients
I've got a problem that looks like $$y = \sum\limits^x_{i=0} { a \choose i } b^i$$ where $a, b$ are constant. I'd like to rewrite this equation to define $x$ in terms of $y$, to directly compute the $...
0
votes
2answers
42 views
The relationship between the Pascal's triangle/sequence and the binomial number theorem?
What is the relationship between Pascal's sequence and the binomial theorem? I want to have a thorough and intuitive understanding of the connections between the two.
Though I am able to relate to ...
1
vote
3answers
65 views
What is the range of convergence of $\sum_{n=0}^{\infty} {(-1)}^n\binom{1/2}{n}\frac{1}{2n+3}.$
I was fiddling with the integral
$$\int_0^1 x^2\sqrt{1-x^2} \ dx $$
and I expanded the term under square root using a binomial series. Integrating, I got the result
$$\sum_{n=0}^{\infty} {(-1)}^n\...
1
vote
1answer
54 views
Prove the following Combinatorial Identity:
I actually would like to try and figure out a proof for myself, but I would like to ask if anyone could provide me a hint to successfully proving the following identity:
$$\sum_{k = 0}^n k \binom{n+1}...
1
vote
1answer
20 views
Sum of independent Poisson distributions and binomial theorem
I have two independent Poisson variables $X_1$ and $X_2$ for which I have calculated the sum in convolution. I obtained:
$$ e ^{\mu_1 \mu_2}\sum_{k=O}\frac{\mu_1^k}{k!}\frac{\mu_z^{z-k_1}}{z-k_1!}$$
...
1
vote
3answers
43 views
Hint on how to prove the identity $\sum_{0 \leq k \leq n } \binom{k}{m} = \binom{n + 1}{m + 1}$
Let $N$ be a positive integer. Then, we have
$\sum\limits_{j=1}^{N} \binom{j}{6} = \binom{N+1}{7}$.
Could anyone explain this equation a little bit? I wrote out the left hand side as $\binom{1}{6} + ...
0
votes
1answer
31 views
Expansion question involving Taylor or Binomial series?
enter image description here
I have tried to solve this question using binomial series, Taylor series but I seem to be heading nowhere… I seem to get lost every time I attempt it.
-1
votes
1answer
65 views
Combinatorics Binomial How to prove it? [closed]
$$
\sum_{k=n}^{2n} \binom{k}{n} 2^{-k}=1
$$
Anyone can help me?
How to prove it?
1
vote
0answers
61 views
Prove the Binomial Theorem using induction (What is mathematics book)
I read What is Mathematics book by Richard Courant and Herbert Robbins and there is an exercise where I should prove the Binomial theorem using induction. Here is an expression to be proved:
$$
C^n_i ...
1
vote
1answer
45 views
How can I find a short term for those sums?
I have $A=[a_1,a_2,...,a_{2016}]$.
The number of subsets of $A$ where the number of elements is divisible by $4$, is $M$.
The number of subsets of $A$ where the number of elements is divisible by $2$ ...
6
votes
3answers
118 views
Proving $\sum\limits_{k=0}^{\infty}\binom {m-r+s}k\binom {n+r-s}{n-k}\binom {r+k}{m+n}=\binom rm\binom sn$
Question: How do you show the following equality holds using binomials$$\sum\limits_{k=0}^{\infty}\binom {m-r+s}k\binom {r+k}{m+n}\binom {n+r-s}{n-k}=\binom rm\binom sn$$
I would like to prove the ...
5
votes
1answer
117 views
How can I prove the following equality?
I need to prove:
$$\sum_{k=0}^{n} {k\choose 2}\cdot{{n-k}\choose 2} = \sum_{k=0}^{n} {k\choose 3}\cdot(n-k).$$
I wasn't able to have any progress with algebra, I tried to think about a combinatorical ...
0
votes
3answers
42 views
Which one is greater (A or B)?
Q. If $$A=(99)^{50} + (100)^{50}$$ and $$ B=(101)^{50}$$ then-
$(a) A>B$
$(b) A<B$
$(c) A=B$
My attempt - I thought of using the binomial approximation to A and B , which gives - $$ A= (100)^...
7
votes
5answers
135 views
How many digits is $99^{99}$ without a calculator?
I know the answer is $198$.
I realise that if $\log _{10}\left( x\right) =y$, the number $x$ has $\lfloor y\rfloor -1$ digits
So I tried $\log ^{\ }_{10}\left( 99^{99}\right) $ = $\log _{10}\left( ...
0
votes
1answer
31 views
Upper bound of of powers of a binomial
I need to proof a property of an object of statistic.
I would have finished successfully if it was not for me to use an upper bound of a power of a binomial that i found in my book that I can't proof ...
0
votes
0answers
24 views
Proving the Hockey Stick Identity with the Summation Identity [duplicate]
The Summation Identity is: $\dbinom{n}{r}+\dbinom{n}{r+1}=\dbinom{n+1}{r+1}$. Is there a way to prove the hockey stick identity with this?
Hockey Stick Identity: $\dbinom{n}{r}+\dbinom{n+1}{r}+\...
0
votes
1answer
40 views
How is $ C_0^n+\frac{1}{2}(C_0^n-C_1^n)+\frac{1}{3}(C_0^n-C_1^n+C_2^n)+.. =C_0^{n-1}+\frac{1}{2}C_1^{n-1}(-1)+\frac{1}{3}C_2^{n-1}(-1)^2… $? [closed]
I saw this result in the solution of a question: $ C_0^n + \frac{1}{2}(C_0^n-C_1^n) + \frac{1}{3}(C_0^n-C_1^n+C_2^n)+... = C_0^{n-1} + \frac{1}{2}C_1^{n-1}(-1) + \frac{1}{3}C_2^{n-1}(-1)^2... = 1/n $. ...
