> [!NOTE] Definition (Univariate Real Power Series About Zero) > Let $(a_{n})_{n\geq 0}$ be a [[Real sequences|sequence of real numbers]]. The power series in $x$ centred at $0$ is given by the [[Series of Real Sequence|series]] $\sum_{n=0}^{\infty} a_{n} x^{n}$where $x\in \mathbb{R}$ is a variable. # Properties By [[Radius of Convergence of Complex Power Series]], ... By [[Continuity of Real Power Series About Zero on Interval of Convergence]], ... By [[Cauchy Product of Univariate Real Power Series About Zero Converges to Product]], .... # Applications **Examples**: > [!Example] Examples > > - [[Taylor's Theorem With Lagrange Remainder for Real Function]]: $f(x) =f(a)+f^{\prime}(a)(x-a)+\frac{1}{2 !}(x-a)^2+\ldots$ > - [[Real Exponential Function]]: $e^x =1+x+\frac{1}{2 !} x^2+\frac{1}{3 !} x^3+\ldots$ > - [[Real Natural Logarithm Function]]: $\ln (1+x) =x-\frac{1}{2} x^2+\frac{1}{3} x^3-\frac{1}{4} x^4+\ldots, \quad|x|<1$ > - [[Trigonometric Functions]]: $\begin{align} > \sin (x) & =x-\frac{1}{3 !} x^3+\frac{1}{5 !} x^5-\frac{1}{7 !} x^7+\ldots \\ > \cos (x) & =1-\frac{1}{2 !} x^2+\frac{1}{4 !} x^4-\frac{1}{6 !} x^6+\ldots \\ > \end{align}$ > - [[Binomial Series]]: $(1+x)^p =1+p x+\frac{p(p-1)}{2 !} x^2+\frac{p(p-1)(p-2)}{3 !} x^3+\ldots, \quad|x|<1$; $(a+x)^p =a^p+p a^{p-1} x+\frac{p(p-1)}{2 !} a^{p-2} x^2+\frac{p(p-1)(p-2)}{3 !} a^{p-3} x^3+\ldots, \quad\left|\frac{x}{a}\right|<1$ > - [[Geometric Series]]: $\sum x^{n}$. >