Questions tagged [binomial-theorem]

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

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Representing the generating function using binomial theorem.

Lets say we have a set {1, 2, 3, 4, 5}, There are 32 subsets. Now if we take these subsets and arrange them according to the sum of their members; For eg: The subset {2, 3} would be in the same group ...
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Prove $(a+b)^{p} \leq a^{p}+b^{p}$ if $a,b>0$ and $p \in (0,1)$ [duplicate]

I need to prove that if $a,b>0$ and $p \in (0,1)$ then $(a+b)^{p} \leq a^{p}+b^{p}$. I've been trying to use the generalized binomial theorem but i haven't solve it yet.
2 votes
1 answer
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Show $\int_0^1 x^n (1-x)^{N-n} dx = \left(\binom{N}{n} (N+1)\right)^{-1}$ [duplicate]

As the title suggests I would like to know how to solve the following integral (solution found w Mathematica) $$\int_0^1 x^n (1-x)^{N-n} dx = \left(\binom{N}{n} (N+1)\right)^{-1}$$ (we may assume $0&...
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1 answer
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When $\text{ord}_n(p)/m = \text{ord}_{n/m}(p)=1$

$\newcommand\ord{\text{ord}}$ For integers $n,m,p$, suppose $m$ and $n$ share the same prime divisors with $m$ dividing $n$, and suppose $\text{gcd}(p,n)=1$. I want to show that $\ord_n(p)/m=\ord_{n/m}...
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quickest way of determining the the coefficient of any term of a binomial expansion

What are the coefficients of $x^4$ and $x^5$ of this binomial $(1+x)\left(1-\frac x2\right)^8$? and also share the quickest way of solving this type of math (without expanding it fully if possible)!
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1 vote
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How to determine the following coefficient?

I have such $$[x^8](1-5x^4)^{-2}(1+3x)^{-6}$$ I simplify as $$[x^8]\sum_{n \geq 0} \binom{n+2-1}{2-1}(5x^4)^n \cdot \sum_{n \geq 0} \binom{n+6-1}{6-1}(-3x)^n $$ $$[x^8]\sum_{n \geq 0} \binom{n+1}{1}5^...
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Prove $\frac{((1+(|k|)^{1/8})^{4n}}{n(n-1)(2n-1)(2n-3)}\leq \frac{(4n-1)(4n-3)(4n-5)(4n-7)}{630}|k|$

Determine whether it is true or false. There exists some $k\in \Bbb R,n\in \Bbb N$ \ {$0,1$} such that $\frac{(1+(|k|)^{1/8})^{4n}}{n(n-1)(2n-1)(2n-3)}\leq \frac{(4n-1)(4n-3)(4n-5)(4n-7)}{630}|k|$. I ...
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How do I solve a binomial theorem with a factor inserted in

I have a given formula which is very similar to the Binomial Formula $$\sum_{k=0}^n \frac{k}{n} \binom{n}{k} z^k (1-z)^{n-k} = z$$ I have to prove that the above statement is true. As of right now I ...
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How to show that this sum is zero, $\sum^{k}_{j=0}\frac{(-1)^{j}}{j!(k-j)!} $ whenever $k \ge 1$?

Is there a simple way to prove this sum, $$S_{k} =\sum^{k}_{j=0}\frac{(-1)^{j}}{j!(k-j)!}~ ,$$ is zero whenever $k \ge 1$? Whenever $k$ is odd, I know I can pair up the terms and it cancels out, but I'...
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Show that $\binom{r}{k}(p-\frac{k}{n})^k(q-\frac{(r-k)}{n})^{r-k} < q_k < \binom{r}{k}p^kq^{r-k}(1-\frac{r}{n})^{-r}$

Show that $$\binom{r}{k}\left(p-\frac{k}{n}\right)^k\left(q-\frac{r-k}{n}\right)^{r-k} < q_k < \binom{r}{k}p^kq^{r-k}\left(1-\frac{r}{n}\right)^{-r}$$ $q_k$ is given by $$\begin{align}q_k&=\...
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1 vote
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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(...
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Arithmetic triangle and choices

In regards to the arithmetic triangle: 1 1 1 1 2 1 etc we know that it is a way to calculate the binomial coefficients or number of choices $C(n,r)$. ...
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Evaluate $\sum_{r=1}^{n}[\sum_{k=1}^{r}k][\log_{1/2}\sqrt{4x-x^2}]^r$

$$ {\rm Evaluate} \sum_{r=1}^{n}\Big(\sum_{k=1}^{r}k\Big)\big(\log_{1/2}\sqrt{4x-x^2}\big)^{r} $$ First step is probably to make it into $S=\sum_{r=1}^{n}[\frac{r(r+1)}{2}]A^r$ where $A=\log_{1/2}\...
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Shifted Schwartz Functions are Schwartz?

I am asking about the same question as this post: Translation of a Schwartz function is a Schwartz function? Namely: If $f\in S(\mathbf{R}^n)$ and for all $y\in\mathbf{R}^n$ then $\tau_{y}f\in S(\...
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2 votes
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Distributing 10 swords among 3 champions.

Right, so the protocol here seems to be stars and bars. Now let's say the blacksmith is to make sure that each champion gets two swords. We may fix two for each champion and count the combinations for ...
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Comparison between $n$ and $\ln (n)$.

Suppose there exists a $n_0>1, \alpha>0$ such that for all $n >n_0$ following is true: $n- \frac{1}{\alpha}\ln (n) > \frac{n}{2}$. How to find the $n_0$?
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1 vote
0 answers
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Why Bezier Curve is not undefined at t=0 and t = 1? [duplicate]

Sorry,this might be a dumb question, but I couldn't really understand it. If we define Bezier Curves as: $B(t) = \displaystyle\sum_{i = 0}^{n} P_i\binom{n}{i}t^i(1-t)^{n-i}$ when t and i are zero it ...
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If $(1+x+x^2)^n=a_0+a_1x+a_2x^2+\dots+a_{2n}x^{2n}$ then prove that $a_0^2-a_1^2+a_2^2-a_3^2+\dots+(-1)^{n-1}a_{n-1}^2=\frac12a_n(1-(-1)^na_n)$

If $(1+x+x^2)^n=a_0+a_1x+a_2x^2+\dots+a_{2n}x^{2n}$ then prove that $a_0^2-a_1^2+a_2^2-a_3^2+\dots+(-1)^{n-1}a_{n-1}^2=\frac12a_n(1-(-1)^na_n)$ My Attempt: Replacing $x$ by $-x$ in the given ...
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2 votes
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The integer next above $(\sqrt3+1)^{2m}$ contains $2^x$ as a factor. Find $x$.

Question The integer next above $(\sqrt3+1)^{2m}$ contains $2^x$ as a factor. Find $x$. Attempt 1 $(\sqrt3+1)^{2m}=(4+2\sqrt3)^m=2^m(2+\sqrt3)^m=I+f$, where $I$ is the integral value and $f$ is the ...
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Is the binomial formula a Taylor series?

Suppose $r, x, y$ is a nonnegative integer, then \begin{aligned}(x+y)^{r}&=\sum _{k=0}^{\infty }{r \choose k}x^{r-k}y^{k}\end{aligned} by binomial theorem. Is it a Taylor series of a bivariate ...
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Solving for $n$.

Setup: Suppose that $n \ {\rm exp}(-n\gamma)< \tau$ where $n \in \mathbb{N}, \tau \in (0,1], \gamma >0$. How to solve for $n$? My attempt: Let $ n\triangleq {\rm ln} \ \alpha$, then we have the ...
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1 vote
2 answers
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Is there a closed-form expression for the first k terms in a binomial series?

This is kind of two questions in one. Firstly, does the following expression have a closed form? $$\sum_{i=0}^k \binom{n}{i}x^i$$ $$\text{(first $k$ terms in binomial series)}$$ where $k$ is some ...
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Is it possible to evaluate $\lim\limits_{x\to \infty}\sqrt{x^2+x}-x$ by factoring out $\sqrt x$ and using binomial expansion?

To find the $\lim\limits_{x\to \infty}\sqrt{x^2+x}-x$ by using binomial expansion, we would first factor out $\sqrt {x^2}=x$ to make expression in the form $(1+p)^n$ like below: $$\lim\limits_{x\to \...
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Given $p$ red balls and $q$ blue balls, in how many ways can I pick 2 red, then 2 blue, then 2 red, in a row?

Arranging $p$ red balls and $q$ blue balls on a single line, in how many ways can I put 2 red, then 2 blue, then 2 red, in a row? Even further, in how many ways can I pick 2 reds 2 blues 2 reds in a ...
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2 votes
1 answer
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An equivalent formula for $\sum_{1\le i_1\lt i_2 \dots \lt i_n\le n} a_{i_1} a_{i_2} \dots a_{i_n}$

I know that the following holds: $\sum_{1\le i\lt j\le n} a_j a_k = \frac12\left(\left(\sum a_i\right)^2-\sum a_i^2\right)$ The question is: does some equivalent formula holds for products of $q>...
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An integral inequality in Rudin's Principles of Mathematical Analysis

In Rudin's Principles of Mathematical Analysis, the proof of Theorem 7.26 includes the inequality $$\int_0^{1/\sqrt{n}} (1 - x^2)^n dx \ge \int_0^{1/\sqrt{n}} (1 - nx^2) dx,$$ where $n$ is a positive ...
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5 votes
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For $n \geq 2$, show that $\sum_{r = 1}^{n} r \sqrt{\binom{n}{r}} < \sqrt{2 ^ {n - 1} n ^ 3}$

Good day, Can someone help me with giving hints for this problem: Show that for $n \geq 2, n \in \mathbb{Z}$, $$\sum_{r = 1}^{n} r \sqrt{\binom{n}{r}} < \sqrt{2 ^ {n - 1} n ^ 3}$$ I tried $$\sum_{...
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4 votes
2 answers
100 views

Proving identity for binomial coefficient using generating funtion

I am trying to show that $$\sum_{k=0}^n (-1)^k \binom{n}{k} \binom{nx-kx}{n+1} = n x^{n-1} \binom{x}{2}.$$ By using $(1+x)^n(1+x)^n = (1+x)^{2n}$ and $(1+x)^n(1-x)^n = (1-x^2)^n$ and generating ...
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1 vote
2 answers
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Proving ${n + 2 \choose m + 1} = {n \choose m + 1} + 2{n \choose m} + {n \choose m - 1}$ using the Binomial Theorem [duplicate]

I want to prove the following identity using the Binomial Theorem: $${n + 2 \choose m + 1} = {n \choose m + 1} + 2{n \choose m} + {n \choose m - 1}$$ I have a combinatorial solution: Suppose we want ...
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3 votes
1 answer
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Show that $\frac{\binom{n}{1}}{1} + \frac{\binom{n}{2}}{2} + \frac{\binom{n}{3}}{3} + \cdots + \frac{\binom{n}{n}}{n} = \sum_{r=1}^{n}\frac{2^r-1}{r}$

Good day, I was solving this problem: Show that $$\frac{\binom{n}{1}}{1} + \frac{\binom{n}{2}}{2} + \frac{\binom{n}{3}}{3} + \cdots + \frac{\binom{n}{n}}{n} = \sum_{r=1}^{n}\frac{2^r-1}{r}$$ I had ...
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3 votes
1 answer
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How to find the number of zeros at the end of this summation

I've found this problem in a math competition: Given two positive integers $a$, $b$, we define $$ f(a,b)=\sum_{\displaystyle \sum_{i=1}^{a}x_i\leq b}\binom{b}{b-x_1}\binom{b-x_1}{b-x_1-x_2}\binom{b-...
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2 votes
1 answer
83 views

Proof by Induction and combination:

According to Binomial Theorem $$ (x+y)^n=\sum_{k=0}^{n}(^{n}_k) x^ky^{n-k} $$ The result is true for n=1, since $$ (x+y)^1=(^{1}_0)y^{1}+(^{1}_1)x $$ Let the result be true for n=m, that is $$ (x+y)^m=...
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1 vote
0 answers
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Finding $n$ in a binomial expansion

The question is about finding $n$, given that $$(5+nx)^2 (1+(3/5)x)^n = 25+100x+\cdots$$ I've tried 2 approaches. Method 1: $$5^2 (1+nx)^2$$ $$25(1+(2)(nx))$$ $$25+50nx$$ And on the other hand... $$(1+...
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1 vote
1 answer
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Why does Negative binomial expansion have infinite terms

Why does negative binomial expansion have infinite number of terms and not equal to the example given below Why is $(x+a)^{-2}$ not equal to $\frac{1}{x^2 + a^2 + 2ax}$? ?
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1 vote
2 answers
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Conditional probability and independence using Bayes' rule and binomial expansion

I would like to start the question by adding some context. Tom is a night owl and is very hungry at night. On day 𝑖, he eats at home with probability 1−𝑝 (0<𝑝<1); or, with probability 𝑝, he ...
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3 votes
1 answer
145 views

What is $\frac{\binom{n}{1}}{1} + \frac{\binom{n}{2}}{2} + \frac{\binom{n}{3}}{3} + \cdots + \frac{\binom{n}{n}}{n}$

Good Day Today, I learnt that $$\frac{\binom{n}{0}}{1} + \frac{\binom{n}{1}}{2} + \frac{\binom{n}{2}}{3} + \cdots + \frac{\binom{n}{n}}{n + 1} = \frac{2 ^ {n + 1} - 1}{n + 1}$$ I changed it a little ...
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0 votes
1 answer
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Formatting of numbers while finding remainder using Binomial theorem

While Dividing $$2^{501}$$ by 21 to find remainder why do we need to write it in the form of $$8(63+1)^{83}$$ and not like $$2(21-5)^{125}$$ like why should it always be in the form of (1+x) ?
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3 votes
1 answer
85 views

Solving the inequality using Binomial Theorem

I had to prove the following inequality $$a^{3/5}b^{2/5}\leq 3a/5+2b/5$$ For real and non negative a and b I was able to prove it using Holder’s inequality and also AM-GM inequality. I need to know if ...
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1 vote
1 answer
72 views

Aymptotic formula/closed form for $ \sum_{r=1}^{n} {n \choose r} \frac{f^{(r-1)}(1)}{(r-1)!}$

For an $f$ infinitely differentiable on $(0,\infty)$ and real valued, consider a finite sum $$a_n= \sum_{r=1}^{n} {n \choose r} \frac{f^{(r-1)}(1)}{(r-1)!}$$ where $f^{(r-1)}(1)=\frac{d^{(r-1)}f(x)}{...
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1 vote
3 answers
115 views

Given $P_n(x)=\frac1{2^n}\left[(x+\sqrt{x^2-1})^n+(x-\sqrt{x^2-1})^n\right]$, prove $P_n(x)-xP_{n-1}(x)+\frac{1}{4}P_{n-2}(x)=0$

Given the expression $$P_n(x)=\dfrac{1}{2^n}\left[\left(x+\sqrt{x^2-1}\right)^n+\left(x-\sqrt{x^2-1}\right)^n \right]$$ Prove that $P_n(x)$ satisfies the identity $$P_n(x)-xP_{n-1}(x)+\frac{1}{4}P_{n-...
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  • 1,563
-1 votes
1 answer
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Show that $\sum_{i=0}^{n}\binom{k}{i}=2^{k-1}$ [duplicate]

Show that $$\sum_{i=0}^{n}\binom{k}{i}=2^{k-1}$$ for every positive odd integer in the form of $k=2n+1$ ($n$ being a positive integer). I tried the usual induction demonstration. The case for $k=1$ is ...
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1 vote
3 answers
68 views

Binomial expansion of $(1+x)^i$ where $i^2 = -1$.

I was reading today about this single variable binomial expansion $(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+\ldots$ For example, $(1+x)^2 = 1+2x+x^2$ However, is it also valid when ...
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  • 805
1 vote
1 answer
71 views

Proving divisibility by induction [duplicate]

The problem consists of demonstrating through induction that $\forall a \in \mathbb{N}$ and $n\geq 0$ (where $n$ is an integer) $$(a^2+a+1)\mid(a^{n+2} +(a+1)^{2n+1})$$ That is, $a^2+a+1$ divides $a^{...
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0 votes
1 answer
55 views

Prove that $(\frac{13}{16})^{12}=\sum_{k=0}^{12}{12 \choose k}\left(\frac{1}{4}\right)^{2k}\left(\frac{3}{4}\right)^{12-k}$ [closed]

I tried to solve a probability question where I got $\sum_{k=0}^{12}{12 \choose k}\left(\frac{1}{4}\right)^{2k}\left(\frac{3}{4}\right)^{12-k}$ but did not succeed in evaluating that to $(\frac{13}{16}...
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  • 3
3 votes
3 answers
66 views

Given that $z=1+i$, find the value of $n\in\mathbb{Z^+}$ such that $z^n$ is real.

Given that $z=1+i$, find the smallest value of $n\in\mathbb{Z^+}$ such that $z^n$ is real. I'm wondering if there's an algebraic way of solving this question, aside from the obvious trial and error ...
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  • 315
0 votes
2 answers
34 views

Elementary number theory problem involving binomial theorem [duplicate]

Exercise. Let $a,b$ be real numbers, such that $a+b$ and $ab$ are integers. Show that $\forall n\in \Bbb N$: $(a^n+b^n)\in \Bbb Z$. I tried to use the binomial formula: $(a+b)^n=\sum_{k=0}^n{{n}\...
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1 vote
1 answer
91 views

Can we re-write Newton's Binomial formula as a power series in $\ r\ $ without any problems?

Newton's Generalised Binomial theorem states that if $\ x\ $ and $\ y\ $ are real numbers with $\ \vert x \vert > \vert y \vert\ (\text{note that } \left\vert \frac{y}{x} \right\vert < 1),\ $ ...
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0 votes
0 answers
36 views

Evaluating $ \binom{n}{0} + 2 \binom{n}{1} + 2^2 \binom{n}{2} + \dots + 2^k \binom{n}{k} + \dots + 2^n \binom{n}{n} $ [duplicate]

With $n$ a positive integer, evaluate the sum $$ \binom{n}{0} + 2 \binom{n}{1} + 2^2 \binom{n}{2} + \dots + 2^k \binom{n}{k} + \dots + 2^n \binom{n}{n} $$ I'm pretty sure that this has to do with the ...
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  • 315
0 votes
1 answer
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Is the series expansion of $\frac1{(1+x)^n}$ same as $\frac1{(1-x)^n}$ with $(-1)^r$

The series expansion for $$\frac1{(1-x)^n} = \sum_{r=0}^{r=\infty}C_r^{|n|+r-1}x^r$$ Is the expansion of $$\frac1{(1+x)^n} = \sum_{r=0}^{r=\infty}(-1)^rC_r^{|n|+r-1}x^r$$ ?? (Where C is combination)
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0 votes
1 answer
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If particular terms of a binomial expansion are given then does it need to be a unique expansion?

We have a binomial expansion $(x+y)^n$ , where $x,y,n$ are all some real numbers and are unknown. (n$\in$ I), (x, y ≠ 0) And we know the numerical values of $(r-2)$th, (r)th and $(r+2)$th terms of the ...
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