query: How to compute $\int_{-\pi}^\pi\bigg(\sum_{n=1}^\infty\frac{\cos(nx)}{2^n}\bigg)^2dx$
positive_passages.docid: 24022
positive_passages.text: Compute $\int _0 ^{2\pi} (\sum_{n=1}^\infty \frac {\cos(nx)}{2^n})^2 dx$
negative_passages.docid: 91784
negative_passages.text: Calculating $\sum_{n=1}^\infty {\frac{1}{2^nn(3n-1)}}$
negative_passages.docid: 67263
negative_passages.text: Compute integral $\int_{-\infty}^\infty \frac{x^4}{1+x^8} \, dx$
negative_passages.docid: 8705
negative_passages.text: Evaluation of $\sum_{x=0}^\infty e^{-x^2}$
negative_passages.docid: 55339
negative_passages.text: Find the sum $\sum_{n=1}^{\infty}\frac{ 1}{n^{2}(2n-1)}$
negative_passages.docid: 137060
negative_passages.text: Evaluating $\int_0^{\tfrac{\pi}{4}}\ln(\cos x-\sin x)\ln(\cos x) dx - \int_0^{\tfrac{\pi}{4}}\ln(\cos x+\sin x)\ln(\sin x)dx=\frac{G\ln 2}{2}$
negative_passages.docid: 130967
negative_passages.text: How to show that $\int_{0}^{\infty}{\sin(x)\sin(2x)\sin(4x)\cdots\sin(2^kx)\over x^{k+1}}\mathrm dx=2^{0.5(k^2-k-2)}\pi?$
negative_passages.docid: 84765
negative_passages.text: Evaluating $\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$
negative_passages.docid: 121337
negative_passages.text: Divergence of $\sum_{n=1}^{\infty} \frac{1}{2+i^n}$?
negative_passages.docid: 72502
negative_passages.text: Evaluate $\sum_{n=1}^\infty nx^{n-1}$
negative_passages.docid: 120085
negative_passages.text: Compute $\int \frac{-3}{2x^2-x-1} dx$
negative_passages.docid: 53658
negative_passages.text: Find the sum series $\sum_{n=1}^{+\infty} \frac{(-1)^{n-1}(2n-1)}{2^{n-1}}$
negative_passages.docid: 60572
negative_passages.text: Computing $\int_{-\infty}^\infty \frac{\sin x}{x} \mathrm{d}x$ with residue calculus
negative_passages.docid: 122843
negative_passages.text: Finding $\int_0^\infty \frac{\cos(ax)-\cos(bx)}{x}dx$
negative_passages.docid: 118800
negative_passages.text: Calculating $\int_0^\pi \log(1-2a\cos (x)+a^2)\,dx$
negative_passages.docid: 49074
negative_passages.text: Compute $\int\limits_{-\infty}^{\infty}\frac{1}{(1+x^2)^{n+1}}dx$ via residue calculus.
negative_passages.docid: 43903
negative_passages.text: Find the sum for $\sum_{n=1}^{\infty}(-1)^{(n+1)}\frac{2n+1}{3^n} $