query: Proving that $\exp(z_1+z_2) = \exp(z_1)\exp(z_2)$ with power series
positive_passages.docid: 101499
positive_passages.text: Prove $e^{x+y}=e^{x}e^{y}$ by using Exponential Series
negative_passages.docid: 87411
negative_passages.text: Prove that $\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \geq (n-1) \left (\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+\cdots+\frac{1}{\sqrt{x_n}} \right )$
negative_passages.docid: 99676
negative_passages.text: Prove $\sum{\frac{1}{(x+2y)^2}} \geq\frac{1}{xy+yz+zx}$
negative_passages.docid: 76311
negative_passages.text: $f'' + f =0$: finding $f$ using power series
negative_passages.docid: 33903
negative_passages.text: Prove that $(1+a_1) \cdot (1+a_2) \cdot \dots \cdot (1+a_n) \geq 2^n$
negative_passages.docid: 78712
negative_passages.text: Prove that $\frac{1}{\sin^2 z } = \sum\limits_{n= -\infty} ^ {+\infty} \frac{1}{(z-\pi n)^2} $
negative_passages.docid: 50972
negative_passages.text: Prove that $\sum_{x=1}^{n} \frac{1}{x (x+1)(x+2)} = \frac{1}{4} - \frac{1}{2 (n+1) (n+2)}$
negative_passages.docid: 114812
negative_passages.text: Let $p(z)=z^n+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+...+a_0$ be a complex polynomial satisfying $|p(z)| \leq 1$ for $|z|\leq 1$, then prove that $p(z)=z^n$
negative_passages.docid: 112080
negative_passages.text: Proving the Product Rule for exponents with the same base
negative_passages.docid: 101792
negative_passages.text: By expanding $e^x$ into a series prove the following inequality
negative_passages.docid: 116014
negative_passages.text: Write $\frac {1}{1+z^2}$ as a power series centered at $z_0=1$
negative_passages.docid: 38062
negative_passages.text: Proving the convergence of $\sum\limits_{n=1}^{\infty}\frac{1}{1+z^n}$ for $|z| > 1$
negative_passages.docid: 148330
negative_passages.text: Induction Proof for $F_{2n} = F^2_{n+1} - F^2_{n-1}$
negative_passages.docid: 105943
negative_passages.text: Laurent series for $1/(e^z-1)$
negative_passages.docid: 109295
negative_passages.text: Proving $1-\exp(-4x^2/ \pi) \ge \text{erf}(x)^2$
negative_passages.docid: 33011
negative_passages.text: Prove that the polynomial $f_n(x)=nx^{n+1}-(n+1)x^n+1$ is divisible by $(x-1)^2$
negative_passages.docid: 92440
negative_passages.text: Let $n \in \mathbb{Z}^+$, prove the identity $ \sum_{k=1}^{n-1} \binom {n} {k} \frac{kn^{n-k}}{k+1}=\frac{n(n^{n}-1)}{n+1}$