query: How to prove that $\sum_{k=1}^m \cos^2(\frac{2 \pi k}{m}) = \frac{m}{2}$ for $m>2$
positive_passages.docid: 143428
positive_passages.text: How to sum $\sum_{x=0}^{N-1} \cos{\left( \frac{2\pi x}{N} \right)}^{2}$?
negative_passages.docid: 22524
negative_passages.text: Proving that $\frac{\pi^{3}}{32}=1-\sum_{k=1}^{\infty}\frac{2k(2k+1)\zeta(2k+2)}{4^{2k+2}}$
negative_passages.docid: 141645
negative_passages.text: Trigonometric series sum proof
negative_passages.docid: 78712
negative_passages.text: Prove that $\frac{1}{\sin^2 z } = \sum\limits_{n= -\infty} ^ {+\infty} \frac{1}{(z-\pi n)^2} $
negative_passages.docid: 92440
negative_passages.text: Let $n \in \mathbb{Z}^+$, prove the identity $ \sum_{k=1}^{n-1} \binom {n} {k} \frac{kn^{n-k}}{k+1}=\frac{n(n^{n}-1)}{n+1}$
negative_passages.docid: 137249
negative_passages.text: find $\sum_{k=2}^{\infty}\frac{1}{k^2-1}$
negative_passages.docid: 71413
negative_passages.text: Proving complex series $1 + \cos\theta + \cos2\theta +... + \cos n\theta $
negative_passages.docid: 32823
negative_passages.text: Prove $\forall n \in \mathbb{N}: \int_{0}^{\frac{\pi}{2}} |\frac{\sin(nx)}{x}|dx \geq \frac{2}{\pi}\sum_{k=1}^{n}\frac{1}{k}$
negative_passages.docid: 10675
negative_passages.text: Proving that $\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{it})|^2dt=\sum_{n=0}^{\infty}|a_n|^2r^{2n}$
negative_passages.docid: 132065
negative_passages.text: Prove sum of $\cos(\pi/11)+\cos(3\pi/11)+...+\cos(9\pi/11)=1/2$ using Euler's formula
negative_passages.docid: 154544
negative_passages.text: Prove that $ \sum_{n=0}^\infty \frac{z^{2^n}}{1-z^{2^{n+1}}}=\frac{z}{1-z} $
negative_passages.docid: 142132
negative_passages.text: How do you prove $\sum \frac {n}{2^n} = 2$?
negative_passages.docid: 104658
negative_passages.text: proving $\csc^2 \left( \frac{\pi}{7}\right)+\csc^2 \left( \frac{2\pi}{7}\right)+\csc^2 \left( \frac{4\pi}{7}\right)=8$
negative_passages.docid: 96412
negative_passages.text: Prove $ \sum \frac{\cos n} { \sqrt n}$ converges
negative_passages.docid: 54604
negative_passages.text: Evaluate $ \sum_{m > n > 0} \frac{m^2 - n^2}{(m^2 + n^2)^2} $
negative_passages.docid: 50972
negative_passages.text: Prove that $\sum_{x=1}^{n} \frac{1}{x (x+1)(x+2)} = \frac{1}{4} - \frac{1}{2 (n+1) (n+2)}$
negative_passages.docid: 60230
negative_passages.text: Prove that $\frac{1+\sin\theta+i\cos\theta}{1+\sin\theta-i\cos\theta}=\sin\theta+i\cos\theta$