Questions tagged [binomial-theorem]

Question 1

I came across this binomial sum- $p+q=99^{98}(1+99^{2})-\frac{99 \cdot 98^{98}}{1}(1+98^{2})+\frac{99 \cdot 98 \cdot 97^{98}}{1 \cdot 2}(1+97^{2})-\frac{99 \cdot 98 \cdot 97\cdot 96^{98}}{1 \cdot 2 \...

Question 2

Evaluate: $$4^9-\binom{8}{1}4^8+\binom{7}{2}4^7-\binom{6}{3}4^6+\binom{5}{4}4^5$$ My Attempt Doing normal arithmetic calculation the value comes out to be $5120$ which is not a big deal. But can it be ...

Question 3

Let $x$ be a real number. Consider the following expression: $$ \sqrt[3]{\frac{x^{3} - 3x + \left(x^{2} - 1\right)\sqrt{x^{2} - 4}}{2}} + \sqrt[3]{\frac{x^{3} - 3x - \left(x^{2} - 1\right)\sqrt{x^{2} -...

Question 4

Consider the following question In the expansion of $(1 + x)(1 - x^2)\left(1 + \dfrac{3}{x} + \dfrac{3}{x^2} + \dfrac{1}{x^3}\right)^5$, $x \ne 0$, the sum of the coefficients of $x^3$ and $x^{-13}$ ...

Question 5

Using multinomial theorem the powers of terms should sum to $a + 2b + 3c + 4d + 5e = 5$; We get following solutions $e = 1$; $a = 1, d = 1$; $b = 1, c = 1$; $c = 1, a = 2$; $a = 1, b = 2$; So we get $\...

Question 6

Using the integral, $$\frac{1}{\binom{n}{r}} = (n+1)\int_0^1 x^r(1-x)^{n-r} \mathrm dx,$$ I want to solve some binomial questions. For instance, it seems, I might be unclear as to what I require. I am ...

Question 7

I am reading this article by Zudilin. In page $523$, there is a formula for $a,b$ odd positive integers, $n$ a positive integer and $\tau$ a complex number, $$\sin^{2m+b}\pi n\tau=\chi_m \frac{(-1)^{(...

Question 8

Prove that $$\lim_{n \to \infty} \sum_{k = 1}^{n} (-1)^{\,k-1}\, \frac{k}{2^{k}-1}\, \binom{n}{k} = \frac{1}{\ln 2}.$$ I got up to this using binomial expansion and its property and using infinite GP ...

Question 9

The binomial formula $(x+y)^{n}=\sum_{k=0}^{n}\binom{n}{k}x^{k}y^{n-k}$ could be interpreted, when $x,y$ are positive integers, as a two-way counting of "words" of length $n$ which use ...

Question 10

Context While doing some calculations I arrived to the sum: $$S=\sum_{n=0}^{\infty}\frac{(2n)!\cos{(\frac{\pi n^2}{4})}}{2^{4n}(1-2n)n!^2}\tag{1}.$$ I tried to decompose the sum in four sums since: $$\...

Question 11

Problem statement For all $x>0$, $n, m \in \mathbb{N}$, let $$ S(x, n, m) \triangleq \sum_{j=0}^n \binom{n}{j} (1 - x)^{n-j} x^j j^3 \frac{1}{1 + \frac{j}{m} } $$ Assuming $n = o(m)$, for all $c&...

Question 12

There are many examples of $1<x<2$ such that the sequence $\left\lfloor x\right\rfloor, \left\lfloor x^2\right\rfloor, \left\lfloor x^3\right\rfloor, \ldots, \left\lfloor x^k\right\rfloor$ is of ...

Question 13

Let $S_n=\sum_{r=0}^n \sum_{s=0}^n \frac{(-1)^{r+s}}{{n \choose r}\left({n \choose s}+ {n \choose r}\right)}.$ Interchange $r$ and $s$, to have $S_n=\sum_{s=0}^n \sum_{r=0}^n \frac{(-1)^{s+r}}{{n \...

Question 14

$$N(a,b)=\sum_{r=1}^{b} (-1)^{b-r}~{b\choose r}~r^a;\quad a,b \in \mathbb{N}\tag{*}$$ gives number of onto maps from $A \to B$, where number of elements in sets $A$ and $B$ are $a$ and $b$, ...

Question 15

Evaluate $\sum_{r=1}^{101}(-1)^{r-1}(1+\frac12+\frac13+...+\frac1r){101\choose r}$ My Attempt: If $r$ was starting from $0$, I could have written $(1-1)^{101}$. Now, maybe ${101\choose r}$ can be ...

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Questions tagged [binomial-theorem]

Binomial sum without using Stirling numbers

Evaluate: $4^9-\binom{8}{1}4^8+\binom{7}{2}4^7-\binom{6}{3}4^6+\binom{5}{4}4^5$

Is it possible to simplify a sum of these two cube roots by completing the cube?

Coefficients of $x^3$ and $x^{-13}$ in multiplication of rational functions.

In the expansion of $(1 + x + x^2 + \cdots + x^{10})^3$, what is the coefficient of: $x^5$ [duplicate]

Using the integral $\frac{1}{\binom{n}{r}} = (n+1)\int_0^1 x^r(1-x)^{n-r}\mathrm dx$

A finite sum formula for $\sin^k z$ for $z\in\mathbb{C}$, $k\in\mathbb{N}$

$ \lim\limits_{n \to \infty} \sum_{k = 1}^{n} (-1)^{\,k-1} \frac{k}{2^{k}-1}\, \binom{n}{k} = \frac{1}{\ln 2}$ [closed]

Combinatorial interpretation of the sign-alternating binomial formula

Does the sum have closed form value: $\sum_{n=0}^{\infty}\frac{(2n)!\cos{(\frac{\pi n^2}{4})}}{2^{4n}(1-2n)n!^2}$?

Asymptotic bound on modified binomial sum

Does there exist $1<x<2$ such that the sequence $\lfloor x\rfloor,\lfloor x^2\rfloor,\ldots,\lfloor x^k\rfloor$ is of the form $1,2,3,4,\ldots,n,n+4?$

Finding $\sum_{r=0}^n \sum_{s=0}^n \frac{(-1)^{r+s}}{{n \choose r}\left({n \choose s}+ {n \choose r}\right)}$

Summing $\sum_{r=1}^{b} (-1)^{b-r}~{b\choose r}~r^a;\quad a,b \in \mathbb{N}$

Evaluate $\sum_{r=1}^{101}(-1)^{r-1}(1+\frac12+\frac13+...+\frac1r){101\choose r}$

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