query: Prove that for $z \in\mathbb C\setminus \{1\},\, 1 + z + z^2 +... + z^n = (1-z)^{n+1}/(1 - z)$ for $n = 1,2,3\ldots$
positive_passages.docid: 153295
positive_passages.text: Algebraic Identity $a^{n}-b^{n} = (a-b) \sum\limits_{k=0}^{n-1} a^{k}b^{n-1-k}$
negative_passages.docid: 5927
negative_passages.text: How to prove $a_1^m + a_2^m + \cdots + a_n^m \geq \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}$
negative_passages.docid: 48081
negative_passages.text: Prove that for all $n \in \mathbb{N}$: $2^{n-1}(a^n + b^n) \ge (a + b)^n$
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: 50655
negative_passages.text: Proof of $f^{-1}(B_{1}\setminus B_{2}) = f^{-1}(B_{1})\setminus f^{-1}(B_{2})$
negative_passages.docid: 128913
negative_passages.text: A proof of $\sum_{i=1}^{n}(-1)^{i-1}\binom{n}{i}\frac{1}{i}=\frac{1}{1}+\frac{1}{2}+....+\frac{1}{n}$
negative_passages.docid: 82636
negative_passages.text: Maybe is right $\frac{n^2 + 1}{4k + 3} \notin \mathbb{Z}, n, k \in \mathbb{N}^{+}$
negative_passages.docid: 123467
negative_passages.text: Prove that $f(z) = e^z$ maps $\mathbb{C} \setminus B $ onto $\mathbb{C} \setminus \{0\}$.
negative_passages.docid: 131754
negative_passages.text: Prove $ \forall c \in \mathbb{R^{+}}, \forall B \in \mathbb{N}, \exists n \in \mathbb{N}, (n \geq B) \wedge (\sqrt{2n} > c \ln(n+2)) $?
negative_passages.docid: 136459
negative_passages.text: Prove ${n - 1 \choose k - 1} + {n - 2 \choose k - 1} + {n - 3 \choose k - 1} + \dots + {k - 1 \choose k - 1} = {n \choose k}$
negative_passages.docid: 27922
negative_passages.text: prove that $\binom{n}{0}^2+\binom{n}{1}^2+\binom{n}{2}^2+\cdots+\binom{n}{n}^2=\binom{2n}{n}$
negative_passages.docid: 36906
negative_passages.text: Prove that $x^n+x^{-n} \in \mathbf{N}$ if $x+\frac1x \in \mathbf{N}$
negative_passages.docid: 70625
negative_passages.text: Prove By Induction $1 + 1/2 + 1/3 + ... + 1/(2^n) ≥ 1 + n/2$
negative_passages.docid: 7466
negative_passages.text: How would I prove that if an integer $n>2$, then $\bar{z}=z^{n-1}$ has $n+1$ solutions
negative_passages.docid: 121874
negative_passages.text: Use Induction to prove that for all $n \in \mathbb{N}, (x^n + \frac{1}{x^n}) \in \mathbb{Z}$ if $x+\frac{1}{x}\in\mathbb{Z}$.
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: 30688
negative_passages.text: Prove that $\frac{x_1}{1+x_2+x_3+\ldots+x_n}+\frac{x_2}{1+x_1+x_3+\ldots+x_n}+\ldots+\frac{x_n}{1+x_1+x_2+\ldots+x_{n-1}}\ge\frac{n}{2n-1}$.