query: Identity involving infinite sum of exponentials
positive_passages.docid: 26113
positive_passages.text: Proving the functional equation $\theta (x) = x^{-\frac{1}{2}} \theta (x^{-1})$ from the Poisson summation formula
negative_passages.docid: 12280
negative_passages.text: A dilogarithm identity?
negative_passages.docid: 16776
negative_passages.text: How to prove $\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2}=+\infty$
negative_passages.docid: 156564
negative_passages.text: Prove the identity $\sum_{{{\underset{k-even}{k=0}}}}^{n}{n \choose k}2^{k}=\frac{3^{n}+(-1)^{n}}{2}$
negative_passages.docid: 101499
negative_passages.text: Prove $e^{x+y}=e^{x}e^{y}$ by using Exponential Series
negative_passages.docid: 9124
negative_passages.text: Find the exact value of the infinite sum $\sum_{n=1}^\infty \big\{\mathrm{e}-\big(1+\frac1n\big)^{n}\big\}$
negative_passages.docid: 158515
negative_passages.text: Identity involving Euler's totient function: $\sum \limits_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}$
negative_passages.docid: 99001
negative_passages.text: General identity for a double summation theorem
negative_passages.docid: 118436
negative_passages.text: Identifying the series $\sum\limits_{k=-\infty}^{\infty} 2^k x^{2^k}$
negative_passages.docid: 10875
negative_passages.text: proof of binomial identity involving double sum
negative_passages.docid: 89771
negative_passages.text: $\frac{1}{s-1}+\sum_{n=1}^{\infty}\left[\frac{1}{n^s}+\frac{1}{s-1}\left(\frac{1}{(n+1)^{s-1}} - \frac{1}{n^{s-1}}\right)\right]$
negative_passages.docid: 124229
negative_passages.text: Simplify $\sum_{k=0}^{\infty} x^ky^{\frac{k(k+1)}{2}}$
negative_passages.docid: 58741
negative_passages.text: Operator Exponential $e^A e^B = e^{A+B}$
negative_passages.docid: 48881
negative_passages.text: Verification the integral identity $\int_0^\pi \ln(1+\alpha\cos(x))dx=\pi\ln\left(\frac{1+\sqrt{1-\alpha^2}}{2}\right)$.
negative_passages.docid: 34496
negative_passages.text: Proof that $e=\sum\limits_{k=0}^{+\infty}\frac{1}{k!}$
negative_passages.docid: 45725
negative_passages.text: Derivation of an identity with beta function.
negative_passages.docid: 104078
negative_passages.text: Calculate the sum of the infinite series $\sum_{n=0}^{\infty} \frac{n}{4^n}$