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

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26 views

Proof of the binomial theorem through Dirichlet convolution?

Here I gave a proof for $\sum_{k=0}^n\binom nk(-1)^k=0$ based on the fact that $\mu*1=\varepsilon$ (the Dirichlet identity). I am wondering if using a similar technique we can prove that ...
4
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1answer
269 views

Methods of finding the difference to the nearest integer?

The question asked to find the "smallest value of n such that $(1+2^{0.5})^n$ is within 10^-9 of a whole number." I'm unsure of the approach to the question. The question was in the chapter of ...
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2answers
35 views

Binomial expansions question

In a physics book the autor make the following expansions, given the fact that $z>>d$ (much greater). However I didn't understand how he manage to get the final expression. ...
0
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1answer
51 views

Binomial theorem proof

I'm working through Richard Hamming's "Methods of Mathematics Applied to Calculus, Probability, and Statistics" on my own. I'm struggling with his proof of the binomial theorem, as summarized below. ...
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2answers
52 views

Solve $\frac{1}{2^\theta}\sum_{k=0}^{\theta} {\theta\choose k} \delta(k)=\theta$ for $\delta$

The following arises in unbiased estimation of a parameter for the binomial distribution, but that information is not needed for solving the question. I tried solving this by manipulating the sum to ...
5
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2answers
113 views

When is $(a+b)^n \equiv a^n+b^n$?

I remember a relation like $(a+b)^n \equiv a^n+b^n$, but I don't remember mod which numbers this is true. Where can I learn more about this?
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2answers
60 views

Finding the Derivative of $\sqrt{x}$

How can I find the derivative of $\sqrt{x}$ using first principle. Specifically I'm having difficulty expanding $\sqrt{x + h}$ or rather $(x + h)^.5$. Is there any generalized formula for the ...
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1answer
23 views

How can one derive the newton's generalised binomial series?

How can one derive the relation $$(x+y)^r = x^r + r x^{r-1} y + \frac{r(r-1)}{2!} x^{r-2} y^2 + \frac{r(r-1)(r-2)}{3!} x^{r-3} y^3 + \cdots.$$ if x and y are real numbers with $|x| > |y|$ and $r$ ...
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2answers
48 views

find the coefficient

If $n$ is an odd natural number, and $\sin(n\theta) = \Sigma_{r=0}^{n} b_r \sin^r\theta$, then find $b_r$ in terms of $n$. I have tried this using trigonometric expansion but unable to find solution ...
1
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1answer
19 views

is this inner product positive-definite?

$$\left \langle u, v \right \rangle = pu_{1}v_{1}+qu_{1}v_{2}+qu_{2}v_{1}+pu_{2}v_{2}\\\text{ for }\\ \text{p >0} \text{ and } p^{2}\geq q^{2}$$ The solution breaks down $$\left \langle u, u ...
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2answers
44 views

Evaluate $\sum_{r=0}^n \binom{n}{r}\sin rx \cos (n-r)x$

Evaluate $$ \sum_{r=0}^n \left[\binom{n}{r}\cdot\sin rx \cdot \cos (n-r)x\right] $$ I tried to use binomial identities, but since there are trigonometric terms, I don't have the idea ...
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2answers
112 views

Binomial expansion.

Find the coefficient of $x$ in the expansion of $\left(2-\frac{4}{x^3}\right)\left(x+\frac{2}{x^2}\right)^6$. I've used the way that my teacher teach me. I've stuck in somewhere else. ...
4
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4answers
144 views

Proof of the Binomial theorem; does ${n-1}\choose{n}$ make sense?

I wanted to read a proof for the Binomial theorem, so I googled "proof of the binomial theorem". My question is about the proof from the top link of that search. In the sixth line of the induction ...
1
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1answer
71 views

Are there expansions of the expression $(a+b)^{1/n}$? [duplicate]

Is there an expansion of the expression in the bracket such as $$ \sqrt{a + b} = (a + b)^{1/2}$$ If not do you know of a method that lets us solve such expression and ones with higher roots?
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3answers
59 views

Evaluate: binomial theorem

Show: $$(x+1)^m=\sum_{k=1}^{m}\binom{m}{k}x^k$$ Can somebody help me in showing the above stated problem?
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0answers
45 views

Why in order to have the greatest term in the expansion of$ (1 + x)^n$, $x$ can't be greater than unity?

I was reading Binomial theorem of any index of Higher Algebra by Hall & Knight; there a section was attributed to find the greatest term in the expansion of $(1+x)^n$ for any rational value of ...
2
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1answer
52 views

Sum of squares of terms of a binomial expansion

I have a coin that show heads with a probability $p$. I toss it $N$ times and count the number of heads. I repeat the experiment once more. What's the probability that I get the same number of heads ...
1
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1answer
62 views

Binomial Theorem with Multiple Items

In the game of Twister, a spinner randomly selects either an arm or a leg, and also selects one of four colors, each with equal probability, and players have to move the appropriate body part to the ...
0
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1answer
44 views

Evaluating a cube root

How to evaluate $(8.024)^{1/3}$ from $(1+3x)^{1/3}$.I already expand it until $x^3$ but i still can't get the answer. I tried googling for the working using binomial theorem but i failed.
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1answer
22 views

How do I properly represent this question? (binomial polynomials)

The total area of a picture and its frame can be represented by (l + 2f)(w + 2f), where l and w represent the length and width of the picture, respectively, and f represents the thickness of the ...
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3answers
52 views

Prove $\lim_{n\rightarrow \infty} n^{\frac{1}{n}} = 1$

Prove that $$\lim_{n\rightarrow \infty} n^{\frac{1}{n}} = 1$$ Hint: write $$n^{\frac{1}{n}} = 1 + \epsilon_n, 0 < \epsilon_n $$ then $$n = (1+\epsilon_n)^n > 1 + \frac{n(n+1)}{2}\epsilon_n^2$$ ...
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1answer
36 views

Substitution in Binomial theorem

By substituting $0.08$ for $x$ in $(1+x)^{1/2}$ and its expansion to find $\sqrt 3$, correct to four significant figure. The answer is $1.732$ given by the practice. I couldn't find the connection by ...
0
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1answer
28 views

Approximating radicals using Binomial theorem

Use a suitable binomial expansion to find square root of $1.01$ and correct it to five decimal places. I use the formula $$(1+ax)= 1+ax + \frac{a(a-1)}{2!} + \cdots$$ but do not know where to stop. ...
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0answers
56 views

Why does this trick for the binomial theorem for negative exponents work?

The "trick" for solving a binomial with a negative index is given by: I understand how to expand binomials of the form $1+x$ for negative indices, but I do not why this method of factoring works ...
0
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1answer
37 views

Bound for non-integer power of sum

Let $x > 1$, $y \in (0,1)$ and $z \in (0,1)$. I need to bound $$(x+y)^z - x^z \leq B_z(x)$$ where I guess something like $B_z(x) \approx x^{z-1}$. Is there anything known on these non-integer ...
0
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1answer
42 views

Multiplication of 2 sums that equal another multiplication of 2 sums

I have been trying to prove a formula of mine and i come across something very interesting, well to me it is. If the formula is correct, it states that: $$ \left(\sum_{m=0}^{k-c} {k-c \choose m}{ms_1 ...
0
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1answer
52 views

Sum on integration and binomial theorem.

If $(1+x)^n = \sum_{r=0}^n \binom{n}{r}x^r$ and $$\sum_{r=0}^n \frac{(-1)^r}{(r+1)^2} \binom{n}{r} = k\sum_{r=0}^n \frac{1}{r+1}$$ Then prove that $$k=\frac{1}{n+1}.$$
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6answers
127 views

What are the last two digits of $77^{17}$?

I'm trying to solve current task referenced the following but I stuck at $\displaystyle77^{17}\equiv x\pmod{100}$. As it is described on above link it uses Binomial Theorem. But I read a lot about the ...
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2answers
27 views

Find the remainder of the following using binomial theorem? [closed]

Find the remainder when $$ 5^{5^{5^{5...}}}$$ (24 times 5 ) is divided by 24 using binomial theorem ? Answer to the question is 5 .
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1answer
88 views

How to find coefficient of $x^{12}$ in the expansion of $(1+x+x^2+x^3+…+x^n)^4$

How to find coefficient of $x^{12}$ in the expansion of $(1+x+x^2+x^3+...+x^n)^4$ I tried this : Since $(1+x+x^2+x^3+...+x^n)$ is in GP its sum will be $(x^{n+1}+1)(x-1)^{-1}$ now ACQ we have to ...
0
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0answers
30 views

Show $\sum_{ r = 0}^{n} {(\binom n r)}^{2} = \frac{(2n)!}{(n!)^{2}}$ [duplicate]

How do we show that this identity holds for any n? Any hints or solutions? Show $\sum_{ r = 0}^{n} {(\binom n r)}^{2} = \frac{(2n)!}{(n!)^{2}}$
2
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1answer
72 views

Prove limit converges in definition of $e.$

I've looked up several related questions, but they do not answer what I am looking for. Please give link if this is a duplicate. What I eventually want to know is why ...
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2answers
55 views

$1+\sqrt[3]{e^{2a}}\sqrt[5]{e^{b}}\sqrt[15]{e^{2c}} \leq \sqrt[3]{(1+e^{a})^2}\sqrt[5]{1+e^{b}}\sqrt[15]{(1+e^{c})^2}$

The inequality $1+\sqrt[3]{e^{2a}}\sqrt[5]{e^{b}}\sqrt[15]{e^{2c}} \leq \sqrt[3]{(1+e^{a})^2}\sqrt[5]{1+e^{b}}\sqrt[15]{(1+e^{c})^2}$ is true for all $a,b,c\in\mathbb{R}$? I've tried to use the ...
2
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1answer
39 views

Binomial expansion in descending power

For example, find, in ascending powers of $x$, the first three terms in the expansion of $(2+5x)^7$. So, $(2+5x)^7=2^7+\binom{7}{1}(2^6)(5x)+\binom{7}{2}(2^5)(5x)^2$. I've no problem to solve this ...
1
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1answer
35 views

prove that $(1+pt)^{p^{r-1}} \equiv 1 \pmod {p^r}$

I need to prove that $(1+pt)^{p^{r-1}} \equiv 1 \pmod {p^r}$ the original question is this: Prove that , any primitive root $r$ of $p^n$ is also a primitive root of $p$ and I'm following the ...
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1answer
69 views

Problem with this challenging summation

I'm having some trouble finding the summation of this series. I tried all I could, but in the end the denominator is creating problem. $$ \sum_{r=0}^{n} (-1)^r ...
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0answers
48 views

Finding a closed form expression for $\sum_{k=\frac {n+2} 2} ^n \binom n k$

Find a closed expression for $\displaystyle\sum_{k=\frac {n+2} 2} ^n \binom n k$, $n$ is even. My attempt: $(1+1)^n = \displaystyle\sum_{k=0} ^ n \binom n k= \sum_{k=0} ^{\frac {n-2} 2}\binom n ...
1
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2answers
19 views

Binomial product expansion

I have seen the following mathematical identity in a book: $$ \prod_{i=1}^{N}\left( 1 + ax_i \right)^c = \left( 1 + \sum_{i=1}^{N}{ax_i} + \cdots + \prod_{i=1}^{N}{a^Nx_i} \right)^c $$ Is this a ...
1
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1answer
32 views

Number-theory proof involving the Binomial Theorem

I am trying to prove the following: if $a,b \in \mathbb{Z}$ then $(a+b)^p \equiv a^p + b^p$ (mod $p$), where $p$ is prime. I am recommended to use the fact that $p \choose k$ is divisible by p for $1 ...
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2answers
61 views

How many ways to distribute $n$ objects into $r$ boxes so that each box have at least $1$ (but no more than $k$) objects?

Example: How many ways are there to distribute 15 fruits to 6 people so that each person has at least 1 fruit but no more than 3? I understand how to do it when we need to make sure that at least ...
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2answers
47 views

Factoring $a^{k} - b^{k}$ [duplicate]

I am a bit lost how to factor $a^{k} - b^{k}$. I know it links to the binomial theorem but I can't remember how to do it. Could anyone explain?
1
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1answer
29 views

The convergence of the binomial expansion

The binomial expansion of $\frac{1}{z-2}$ is $-\frac{1}{2}\sum^\infty_{n=0} (-1)^n \left(\frac{z}{2}\right)$. Does this converge for $\left|\frac{z}{2}\right|<2$? Does $\frac{1}{z} ...
1
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1answer
32 views

Simplifying the binomial expansion

I have $$-\frac{1}{2} \left[1-\frac{z}{2}+\left(\frac{z}{2}\right)^2 -\left(\frac{z}{2}\right)^3+...\right]$$ Why does this equal $-\frac{1}{2} \sum^\infty_{n=0} \left(\frac{z}{2}\right)^n$ and not ...
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4answers
410 views

Can you show me a good approach for taking the limit of this function?

I tried to use binomial expansion, but I didn't get the same result. I would like to know how to approach this please. I know the answer is $\sqrt{e}$. My problem is : $$\lim\limits_{x\to 0} ...
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0answers
31 views

Convergence of a binomial expansion

I have the binomial expansion $$1+(-1)(-2z)+\frac{(-1)(-2)(-2z)^2}{2!}+\frac{(-1)(-2)(-3)(-2z)^3}{3!}+...$$ Which simplifies down to $$1+2z+(-2z)^2+(-2z)^3$$ How do I find out if this binomial ...
5
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3answers
83 views

binomial expression of a powered term [duplicate]

One answer to a previous question of mine asserted that $$k^2=\binom k2+\binom {k+1}2.$$ I checked that the formula is true. However, it intrigued me. Is there a similar expression for $k^3$? How ...
2
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1answer
27 views

Why can the function $f(x)=||A\vec{x}-\vec{b}||^2$ be rewritten as $\vec{x}^tA^tA\vec{x}−\vec{x}^tA^t\vec{b}−\vec{b}^tA\vec{x}+||\vec{b}||^2$

Someone answered a question introducing this transformation of the function $f(x)=||A\vec{x}-\vec{b}||^2$ ; but I cannot get the idea why and how. Looks a bit like a binomial expansion, but I can't ...
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1answer
36 views

Solving equation with many exponents for same variable

$1.32=5(r + r^4) + 10 (r^2 + r^3) + r^5 + 1$ How would one solve an equation like this? Any help would be appreciated thanks.
4
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0answers
43 views

Binomial expansion with negative power?

I tried searching everything but couldn't understand what formula or technique are they using to expand this term. I have attached image of my textbook for this problem. It's written there that we ...
1
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1answer
34 views

Falling power of a sum in terms of falling powers of the terms

I am trying to come up with an expression for $(x+y)^{\underline{n}}$ in terms of $x^{\underline{r}}$ and $y^{\underline{r}}$. I tried for $n=2$ and $n=3$ and it looks like binomial expansion holds, ...