All Questions
9
questions
3
votes
1
answer
215
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Sum with Binomial Coefficients and Sine; $S=\sum_{k=0}^n \binom{n}{k} \sin(kx)$
Sum with Binomial Coefficients
Let $n ∈ ℕ₀$ and $x ∈ ℝ$.
$$S=\sum_{k=0}^n \binom{n}{k} \sin(kx)$$
Simplify the sum to a polynomial in n.
I tried to use Euler's Formula and the Binomial Theorem, ...
- 77
1
vote
2
answers
66
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Closed form for $\sum_{r=0}^{n}(-1)^r {3n+1 \choose 3r+1} \cos ^{3n-3r}(x)\sin^{3r+1} (x)$
I request a closed form for the following sum $$S(n)=\sum_{r=0}^{n}(-1)^r {3n+1 \choose 3r+1} \cos ^{3n-3r}(x)\sin^{3r+1} (x) $$
I tried using De Moivre's theorem $$\cos(nx)+i\sin(nx)=(\cos (x)+i\sin(...
user1103841
1
vote
1
answer
109
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Is it possible to derive the nth derivative of$~\exp\left(x\right)\sin^{}\left(x\right)~$using binomial coefficient$~{n\choose k}~$?
I assume$~n\in\mathbb{N}_{\geq0}~$is held.
$$y=\exp\left(x\right)\sin^{}\left(x\right)$$
$$\left(f\cdot g\right)^{\left(n\right)}=\sum_{k=0}^{n}{n\choose k}g^{\left(k\right)}f^{\left(n-k\right)}$$
\...
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0
votes
2
answers
200
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How to prove $\sum^n_{k=0}\binom{n}k\cos\big((n-2k)\theta\big)=2^n\cos^n\theta$?
Given that $n\in\mathbb{Z}$, for any $\theta\in\mathbb{R}$, prove that
$$\sum^n_{k=0}\binom{n}k\cos\big((n-2k)\theta\big)=2^n\cos^n\theta\,.$$
I tried to finish the proof by Mathematical Induction. ...
- 323
-1
votes
1
answer
61
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Combinatory Analisis and Newton's Binomial [closed]
Why is $C_{n}^{0} \ + \ C_{n}^{3} \ +\ C_{n}^{6}\ +\ ...=\ \frac13\cdot[2^n + 2\cdot\cos({\frac{n\pi}3})]$?
8
votes
2
answers
220
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Proving that ${n+3\choose 3} =\frac{n+2}{2}\sum_{k=1}^{n+1}\csc^2\frac{k\pi}{n+2}$
Fancy physics predicts the equality
$${n+3\choose 3} =\frac{n+2}{2}\;\sum_{k=1}^{n+1}\csc^2\frac{k\pi}{n+2}$$
which I can check (numerically and symbolically) for small $n$, but cannot prove for every ...
- 300
2
votes
0
answers
157
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Rewrite binomial sum with trig functions
From the multisections of sums section in wiki page on binomials, I found the following identity where for $t, s$, $0 \le t \lt s$
$$\tag{Ramus' identity} \sum_{k}{\binom{n}{t + ks}} = \frac{1}{s} \...
- 1,034
11
votes
0
answers
441
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Tricky Sum involving Binomial Coefficients and Sine
I am stumped by the sum
$$\sum_{x=0}^n \binom{n}{x}\sin\big(\frac{\pi x}{n}\big)$$
but I can't figure it out. I tried expanding the taylor series of sine and using Euler's identity, but to no avail. ...
4
votes
1
answer
228
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Prove that $\displaystyle \sum_{1\leq k<j\leq n} \tan^2\left(\frac{k\pi}{2n+1}\right)\tan^2\left(\frac{j\pi}{2n+1}\right)=\binom{2n+1}{4} $
Prove that
$$\sum_{1\leq k < j\leq n}\tan^2\left(\frac{k\pi}{2n+1}\right)\tan^2\left(\frac{j\pi}{2n+1}\right)=\binom{2n+1}{4}$$
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