query: Is $\sum_{n\ge0}\frac1{2^{2^n}}$ rational?
positive_passages.docid: 152227
positive_passages.text: How to sum this series to infinity: $\sum_{n=0}^{\infty} \frac1{2^{2^n}}$
negative_passages.docid: 145274
negative_passages.text: Prove by induction $\sum \frac {1}{2^n} < 1$
negative_passages.docid: 49851
negative_passages.text: when the sum of some fractions be $1$
negative_passages.docid: 144078
negative_passages.text: What does $\sum_{k=0}^\infty \frac{k}{2^k}$ converge to?
negative_passages.docid: 5817
negative_passages.text: Does the series $\sum \limits_{n=2}^{\infty }\frac{n^{\log n}}{(\log n)^{n}}$ converge?
negative_passages.docid: 56464
negative_passages.text: Proving the irrationality of $e^n$
negative_passages.docid: 132984
negative_passages.text: $\sum\limits_{k=0}^{\frac{p-1}2}3^k\binom{p}{k}\equiv 2^p-1\pmod{p^2}$
negative_passages.docid: 86497
negative_passages.text: Prove that any rational can be expressed in the form $\sum\limits_{k=1}^n{\frac{1}{a_k}}$, $a_k\in\mathbb N^*$
negative_passages.docid: 28254
negative_passages.text: Why $\sum_{k=1}^{\infty} \frac{k}{2^k} = 2$?
negative_passages.docid: 125148
negative_passages.text: Summation of series $\sum_{k=0}^\infty 2^k/\binom{2k+1}{k}$
negative_passages.docid: 121554
negative_passages.text: Finding $\sum\frac{1}{2^n(n+1)}$
negative_passages.docid: 110695
negative_passages.text: Divisor of $2^{2^n} + 1$ modulus $2^{n+1}$
negative_passages.docid: 132351
negative_passages.text: Is the infinite sum of $2 ^{-n}$ convergent? Why?
negative_passages.docid: 62504
negative_passages.text: Is there any known relationship between $\sum_{i=1}^{n} f(i)$ and $\sum_{i=1}^{n} \dfrac {1}{f(i)}$
negative_passages.docid: 137997
negative_passages.text: Does the series $\sum \frac{x^n}{1+x^n}$ converge uniformly on $x\in [0,1)$?
negative_passages.docid: 19254
negative_passages.text: How many rationals of the form $\large \frac{2^n+1}{n^2}$ are integers?
negative_passages.docid: 78542
negative_passages.text: What is $\sum_{n=1}^{\infty} \frac{n}{2^{\sqrt{n}}}$?