query: Proving a result of $\text{Re}\left(\frac{1-e^{i(n+1)\theta}}{1-e^{i\theta}}\right)$
positive_passages.docid: 71413
positive_passages.text: Proving complex series $1 + \cos\theta + \cos2\theta +... + \cos n\theta $
negative_passages.docid: 96961
negative_passages.text: A proof for the identity $\sum_{n=r}^{\infty} {n \choose r}^{-1} = \frac{r}{r-1}$
negative_passages.docid: 106609
negative_passages.text: Show that $e^{iy} = 1 + iy + \frac{\mu_1 y^2}{2}$ for all $y \in \mathbb{R}$ with $|\mu_1| \leq 1$
negative_passages.docid: 15164
negative_passages.text: Proof of $\sum^{2N}_{n=1} \frac{(-1)^{n-1}}{n} = \sum^{N}_{n=1} \frac{1}{N+n}$
negative_passages.docid: 114031
negative_passages.text: Show that $\sum_{n=0}^\infty r^n e^{i n \theta} = \frac{1- r\cos(\theta)+i r \sin(\theta)}{1+r^2-2r\cos(\theta)}$
negative_passages.docid: 78033
negative_passages.text: Evaluating $\prod_{n=1}^{\infty}\left(1+\frac{1}{2^n}\right)$
negative_passages.docid: 114411
negative_passages.text: Simple Proof of the Euler Identity $\exp{i\theta}=\cos{\theta}+i\sin{\theta}$
negative_passages.docid: 52353
negative_passages.text: Bounding Euler's e by showing $2 \leq \left (1+ \frac{1}{n}\right)^n < 3$
negative_passages.docid: 157068
negative_passages.text: Prove that $\left (\frac{1}{a}+1 \right)\left (\frac{1}{b}+1 \right)\left (\frac{1}{c}+1 \right) \geq 64.$
negative_passages.docid: 153226
negative_passages.text: Proof of $\sum_{k = 1}^{n} \frac{1}{k^{2}} < 2 - \frac{1}{n}$
negative_passages.docid: 48503
negative_passages.text: Proving : $ \bigl(1+\frac{1}{n+1}\bigr)^{n+1} \gt (1+\frac{1}{n})^{n} $
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: 156766
negative_passages.text: Prove that $\sqrt{\frac{n}{n+1}}\notin \mathbb{Q}$
negative_passages.docid: 26113
negative_passages.text: Proving the functional equation $\theta (x) = x^{-\frac{1}{2}} \theta (x^{-1})$ from the Poisson summation formula
negative_passages.docid: 34767
negative_passages.text: Proving $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)<3$ for all positive integers $n$
negative_passages.docid: 25921
negative_passages.text: Show that $\left(1+\dfrac{1}{n}\right)^n$ is monotonically increasing
negative_passages.docid: 136652
negative_passages.text: Closed form expression for the product $\prod\limits_{k=1}^{n}\left(1 - \frac{1}{ak}\right)$