For questions related to the binomial theorem, which describes the algebraic expansion of powers of binomials.

learn more… | top users | synonyms

1
vote
1answer
41 views

Find a closed formula for $\sum_{n=1}^\infty nx^{n-1}$ [duplicate]

Find a closed formula for $\sum_{n=1}^\infty nx^{n-1}$ I am trying to use the derivative of generalized binomial theorem, $\frac{d}{dx}[(x+1)^r=\sum_{n=0}^\infty \binom{r}{n}x^n] ...
2
votes
0answers
42 views

upper bounding alternating binomial sums

So we know that $\sum_{i=0}^t\binom{m}{i}\binom{n-m}{t-i}=\binom{n}{t}$ by a simple counting argument. Now is there any bound on the quantity $\sum_{i=0}^t(-1)^i\binom{m}{i}\binom{n-m}{t-i}$? Can we ...
0
votes
1answer
29 views

Binomial Theorem Questions [on hold]

If $x+\frac1x=n$, express $x^5+\frac 1{x^5}$ in terms of $n$. In the binomial expansion of $(a-b)^n$, $n$ is greater than or equal to $5$, the sum of the fifth and sixth terms is zero. Determine the ...
1
vote
2answers
54 views

Proof $\dbinom{n}{0} - \frac{1}{2}\dbinom{n}{1} + \cdots + (-1)^n\frac{1}{2^{n-1}}\dbinom{n}{n-1}$

For all $n \ge 1$, $$\binom{n}{0} - \frac{1}{2}\binom{n}{1} + \frac{1}{2^2}\binom{n}{2} - \frac{1}{2^3}\binom{n}{3} + \cdots + (-1)^n\frac{1}{2^{n-1}}\binom{n}{n-1} = 0,$$ if $n$ is even               ...
1
vote
5answers
64 views

Proof $x$, $1+nx≤ (1+x)^n$ [on hold]

Prof using the binomial theorem: for all integers $n ≥0$ and for all nonnegative real numbers $x$, $1+nx ≤(1+x)^n$. Don't have a idea to start this one. I don't know how to use math induction yet, ...
1
vote
1answer
23 views

Summation of series with binomial coefficients

The value of $$\sum {n\choose n-r} (n-r) \sin(r\cdot \pi/n)$$ where $r\in (0 ..,n)$ is equal to? I think the question can be solved by writing the series in reverse order but I am not able to solve ...
0
votes
2answers
66 views

How do I calculate $\sum_{k=1}^{33}\binom{33}{k} k$

I started studying about binom's and sums, How do I calculate $$\sum_{k=0}^{33}\binom{33}{k} k$$ Note: I do know that it is $\binom{33}0\cdot0 + \binom{33}1 \cdot 1 + ... + \binom{33}{33} \cdot 33$, ...
2
votes
0answers
41 views

Expanding trigonometric functions with binomial expansion

I was challenged to take $\cos^{\pi}(\pi)$ and expand it using binomial expansion and $\cos(x)=\frac{e^{xi}+e^{-xi}}2$, which I tried: $$\cos^{\pi}(\pi)=\left(\frac{e^{\pi i}+e^{-\pi ...
1
vote
2answers
28 views

Range of the expression $n^n (\frac{n+1}{2})^{2n}$

Prove that: $n^n (\frac{n+1}{2})^{2n}$ Greater than or equal to $(\frac{n+1}{2})^3$ Greater than or equal to $(n!)^3$ Here $n\in \mathbb{N}$ How to prove this? It seems too complicated. Please ...
-2
votes
1answer
35 views

How to get from left to right-hand side of the equation? $ \sum_{k=0}^{d} \binom{2d+1}{k} = \frac{1}{2} \cdot 2^{2d+1} $

I would like to know how the left hand side of the equation is achieved. In particular why the $\frac{1}{2}$ is there. I don't understand how one can get from the left to the right side. $$ ...
1
vote
1answer
30 views

Binomial expansion in the form $(1+x^2)^n$

I'm used to dealing with binomial expansion in the form $(1+x)^n$. I understand that if the number is not $1$ then you have to divide the whole bracket by something which would make it $1$. However ...
0
votes
1answer
33 views

Use induction and Pascal's Identity to show that $\sum_{k=0}^{r}C(n+k,k) = C(n+r+1,r)$

I know Pascal's Identity is ${n \choose k}={n-1 \choose k-1}{n-1 \choose k}$, but I am not sure how to set up and use the proof to show that $\sum_{k=0}^{r}C(n+k,k) = C(n+r+1,r)$. Can anyone help me ...
0
votes
1answer
51 views

What is the coefficient of ${x}^{101}{y}^{99}$ in the expression of $(2x-3y)^{200}$

I know that I have to use the binomial theorem. So, in following the formula of ${(1+x)}^{n} = {n \choose 0}+{n \choose 1}{x}+{n \choose 2}{x}^{2}...+{n \choose k}{x}^{k}+...+{n \choose n}{x}^{n}$, I ...
8
votes
1answer
104 views

Solve: $x = (x-\frac{1}{x}) ^ {1/9} + (1-\frac{1}{x})^{1/9}$

Solve: $$x = \left(x-\frac{1}{x}\right) ^ {1/9} + \left(1-\frac{1}{x}\right)^{1/9}$$ Simplifying, $$x^{10/9} = (x^2-1)^{1/9}+(x-1)^{1/9}$$ I don't know how to start. Any hint will be helpful.
1
vote
2answers
85 views

Using binomial theorem to prove $a | b^n \Rightarrow a | b$. ( | is divides, a prime, n integer > 1)

I tried expanding $(b-a+a)^n=$[$(b-a)+a$]$^n$ but it just seemed to further complicate the problem. I also tried to prove the contrapositive but that doesn't seem to lead to anywhere to. Is there any ...
0
votes
1answer
33 views

Expansion of $S_n$ is less than $T_n$ [duplicate]

Let $S_n=\{(1+\frac{1}{n})^n\}$ and $T_n=\{1+\frac{1}{1!}+\frac{1}{2!}+\cdots+\frac{1}{n!}\}$ I am trying to prove that $\lim_{n\rightarrow \infty} S_n > \lim_{n\rightarrow \infty} T_n$ Now I ...
1
vote
2answers
41 views

Multinomial coefficient

$n_{1},...,n_{m}$ are nonnegative integers $n = n_{1}+...+n_{m}$ $W[n_{1},...,n_{m}]$ denotes the # of ways to place $n$ balls (of labels $1, 2, . . . , n$) into $m$ boxes $B_{1},...,B_{m}$ such ...
0
votes
1answer
18 views

Stuck on AP using binomial expansion

If the coefficients of $x^{r-1},x^r,x^{r+1}$ in the binomial expansion of $(1+x)^n$ are in AP,prove $n^2-n(4r+1)+4r^2-2=0$ How to start?
1
vote
1answer
25 views

Binomial distribution, explanation formula

I have a really simple question. I can't figure out the meaning of the binomial coefficient in the case of a binomial distribution formula. I know what the formula means, and how to use it for the ...
0
votes
1answer
50 views

Binominal expression simplification

I need to simplify the expression $$\sum_{k = 1}^{10} k\binom{10}{k}\binom{20}{10 - k}$$ Thank you.
1
vote
0answers
21 views

Binomial Expansion on $(\frac{\rho}{r})^p$+$(\frac{z}{h})^p$ $=1$

Good day all! I have a quick question about this problem I ran into: Think of $z$ as a function of $ \rho $. Solve the following equation for z and use binomial expansion to determine how $ z $ ...
-1
votes
3answers
43 views

Reducing this binomial expression [closed]

I need help for showing that: $$\sum\limits_{k=2}^{50} = k \cdot(k-1)\binom{50}{k}$$ is equal to: $$50\cdot 49\cdot 2^{48}$$ please help , thank you.
2
votes
4answers
93 views

Probability involving Binomial Summation

The problem statement is : $A$ has $n$ coins and $B$ has $n+1$ coins. They toss their coins simultaneously. If $p$ be the probability that $B$ will have more heads than $A$ the find $p=?$ One way ...
0
votes
4answers
30 views

How to evaluate this mixed function limit

$$\lim_{x\rightarrow 0}\frac{\sqrt{1+2x}-\sqrt{1-2x}}{\sin x}$$ What I did was use binomial theorem and the fact that $\lim_{x\rightarrow 0}\dfrac{\sin x}{x} = 1$. $$\lim_{x \rightarrow ...
4
votes
4answers
111 views

Show that $2^n-(n-1)2^{n-2}+\frac{(n-2)(n-3)}{2!}2^{n-4}-…=n+1$

If n is a positive integer I need to show that $2^n-(n-1)2^{n-2}+\frac{(n-2)(n-3)}{2!}2^{n-4}-...=n+1$ My guess: Somehow I need two equivalent binomial expression whose coefficients I need ...
0
votes
3answers
49 views

How to show that $i^m-i(i-1)^m+\frac{i(i-1)}{1.2} (i-2)^m-…(-1)^{i-1}.i.1^m=0$?

How to show the following? $$i^m-i(i-1)^m+\frac{i(i-1)}{1.2} (i-2)^m-...(-1)^{i-1}.i.1^m=0$$ (if $i>m$) This seems really complicated.Can't spot any pattern as such :\ .Someone help me out! ...
1
vote
2answers
42 views

Prove that $n^k<(1+a)^n$ for sufficiently large $n$

Prove using the binomial theorem, that for natural numbers $n$ and $k$ and real positive number $a$, $n^k<(1+a)^n$ for all $n>N$. Using the binomial theorem, ...
1
vote
1answer
77 views

How can I evaluate $\sum_{i=0}^\infty \frac{1}{k^i} \binom{2i}{i}$

Evaluate $$\sum_{i=0}^\infty \left(\frac{\binom{2i}{i}}{k^i}\right),$$ where $k$ is a whole number. I can't figure out how to approach this question, as no binomial series has such coefficients.
1
vote
4answers
36 views

How can I find the coefficient of x when the power is greater than the powers of 2 brackets using binomial expansion?

I have been given this question: Find the coefficient of $x^{13}$ in the expansion of $(1 + 2x)^4(2 + x)^{10}$. I know how I would find $x^4$ or lower degrees, but I am unsure how to approach this, ...
1
vote
1answer
38 views

Binomial series expansion when $a=-1$

Maybe this is simple but i couldn't get the idea. For $\left|x/a\right|<1$, the binomial series expansion is \begin{equation} \left(x+a\right)^{t}=\sum_{j=1}^{\infty}\tbinom{t}{j}x^{j}a^{t-j} ...
7
votes
3answers
95 views

Prove the identity $\binom{2n+1}{0} + \binom{2n+1}{1} + \cdots + \binom{2n+1}{n} = 4^n$

I've worked out a proof, but I was wondering about alternate, possibly more elegant ways to prove the statement. This is my (hopefully correct) proof: Starting from the identity $2^m = \sum_{k=0}^m ...
0
votes
1answer
21 views

Closed form of this binomial expression?

Does a closed form for this binomial expression exists? $\sum_{K=2}^{N}\binom{N}{K}P^{K}(1-P)^{N-K}$ Thank you.
0
votes
1answer
30 views

Can this be proved using definite integrals [duplicate]

It's a problem from a high school math book that I've been unable to solve: Prove using definite integrals that, $${n \choose 1}-\frac{1}{2}{n \choose 2}+\frac{1}{3}{n \choose ...
1
vote
1answer
39 views

How to show $\phantom{d}_d C_0+\phantom{d}_d C_4 + \cdots = 2^{d-2} + 2^{\frac{d}{2}-1} \cos(\frac{d \pi}{4}) $?

I want to show following identities \begin{align} &\phantom{d}_d C_0+\phantom{d}_d C_4 + \cdots = 2^{d-2} + 2^{\frac{d}{2}-1} \cos(\frac{d \pi}{4}) \\ &\phantom{d}_d C_1+\phantom{d}_d C_5 ...
1
vote
3answers
49 views

Binomial Theorem: does $\sum\limits_{i=0}^{n}{{n}\choose{i}} = \dfrac{1}{2}\sum\limits_{i=0}^{n+1}{{n+1}\choose{i}}$?

I am reading through an argument that I found somewhere and I see that it assumes that: $$\sum\limits_{i=0}^{n}{{n}\choose{i}} = \dfrac{1}{2}\sum\limits_{i=0}^{n+1}{{n+1}\choose{i}}$$ Intuitively, ...
2
votes
1answer
27 views

Generalisation of Binomial Theorem, Leibniz Formula and similar theorems

Since the beginning of the year, our maths teacher showed us the Binomial Theorem in $\mathbb{R}$\, then in $\mathbb{C}$\, in $M_n(\mathbb{K)}$ with two matrices which commute, and now the Leibniz ...
1
vote
0answers
49 views

Binomial theorem for differences

Let $\Delta x_ {k} = x_{k} - x_{k+1} $ and the convention $\Delta ^{1} = \Delta$ where $x_ {k}$ is a sequence. So $\Delta^{2} x_ {k} = \Delta ( \Delta x_{k} ) = \Delta (x_{k} -x_{k+1}) $ $= x_{k} ...
4
votes
2answers
96 views

Finding the infinite series: $\sum_{m=0}^\infty\sum_{n=0}^\infty\frac{m!\:n!}{(m+n+2)!}$

Evaluating $$\sum_{m=0}^\infty \sum_{n=0}^\infty\frac{m!n!}{(m+n+2)!}$$ involving binomial coefficients. My attempt: $$\frac{1}{(m+1)(n+1)}\sum_{m=0}^\infty ...
-6
votes
1answer
32 views

How to find the value of n? [closed]

It is known that the binomial coefficient of the third term of expanding $(2-x)^n$ is twice the value of $n$. Deduce the value of $n$.
0
votes
3answers
101 views

A sum related to binomial theorem

If $\dfrac{x^2+x+1}{1-x} = a_0+a_1x+a_2x^2+\cdots$ then $\displaystyle\sum_{\gamma = 1}^{50}a_{\gamma} = ??$ Original Image This is a sum related to evaluating a series, from the chapter ...
7
votes
5answers
166 views

Calculate $\lim_{n \to \infty}\binom{2n}{n}$ without using L'Hôpital's rule.

Questions: (1) Calculate $$\lim_{n \to \infty}\binom{2n}{n}$$ (2) Calculate $$\lim_{n \to \infty}\binom{2n}{n} 2^{-n}$$ without using L'Hôpital's rule. Attempted answers: (1) Here I start by ...
3
votes
1answer
49 views

Confusion regarding a series

I tried much but was unable to find the answer. $$f(x) = \frac{1}{3} + \frac{1 \cdot 3}{3\cdot 6} + \frac{1\cdot 3\cdot 5}{3\cdot 6\cdot 9} + \frac{1\cdot 3\cdot 5\cdot 7}{3\cdot 6\cdot 9\cdot 12} ...
8
votes
5answers
355 views

An interesting Sum involving Binomial Coefficients

How would you evaluate $$\sum _{ k=1 }^{ n } k\left( \begin{matrix} 2n \\ n+k \end{matrix} \right) $$ I tried using Vandermonde identity but I can't seem to nail it down.
2
votes
2answers
76 views

Binomial expansion for $(x+a)^n$ for non-integer n

I finally figured out that you could differentiate $x^n$ and get $nx^{n-1}$ using the derivative quotient, but that required doing binomial expansion for non-integer values. The most I can find with ...
3
votes
2answers
75 views

Is there a name for a binomial expansion without coefficients?

I am investigating a problem from George E. Andrews Number Theory (Dover, 1971), discussed previously here: Induction Proof that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1})$ I was led ...
0
votes
1answer
24 views

Find constant of this inequality $\textbf{t}' \textbf{E}^b \textbf{t} \geq (\textbf{t}'\textbf{E} \textbf{t})^a$

Let a vector $\textbf{t} \in \mathbb{R}^2$, $\textbf{E} \in \mathbb{R}^{2 \times 2}$ positive semidefinite matrix. Given that $$ \textbf{t}' \textbf{E}^b \textbf{t} \geq (\textbf{t}'\textbf{E} ...
1
vote
1answer
35 views

Compute $\sum_{k=4}^{n=30} \frac{C(25,k-3) * 4^{2k-7}}{k-2}$

Compute $\sum_{k=4}^{n=30} \frac{C(25,k-3) * 4^{2k-7}}{k-2}$ I started by saying $t = k-3$, so $k=t+3$ then I got 4$\sum_{k=1}^{n=27} C(25,t) * \frac{4^t}{t+1}$ and then $\sum_{k=4}^{n=27} ...
2
votes
3answers
97 views

How to apply the binomial theorem to $(a^n - b^n)/(a - b)$?

I need to know if the binomial theorem can somehow be applied to: $$\frac{a^n - b^n}{a- b}$$ I've done a bit of research but I still don't know where to begin with this one so I can't offer any ...
2
votes
1answer
25 views

Compute $\sum_{k=-1}^{n=24}C(25,k+2)k2^k$

Compute $\sum_{k=-1}^{n=24}C(25,k+2)k2^k$ Well, I've found a solution for it, but I don't understand the line in the orange rectangle, can anyone exlain it please?
1
vote
0answers
104 views

Can be justified $\zeta(3)=\lim_{n\to\infty}-3\sum_{k=1}^n\sum_{\nu=0}^\infty\frac{(-1)^{\nu+1}}{\nu+1}\binom{3k^3-1}{\nu}$?

My main goal is understand useful facts about my computations, that could be wrong, the way shold be too a street without exit looking for a evaluation of Apéry's constant, since I don't use any ...