Third degree Taylor series of $f(x) = e^x \cos{x} $

Taylor's theorem allows you to use the Big O notation: $$\cos(x)= 1-\frac{x^2}{2!}+O(x^4)\quad\mbox{and}\quad e^x=1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}+O(x^4).$$ Therefore $$e^x\cos(x)=1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}+O(x^4)-\frac{x^2}{2!}(1+x+O(x^2))=1+x-\frac{x^3}{3}+O(x^4).$$

Tags:

Sequences And Series

Real Analysis

Power Series

Taylor Expansion

Related

A curious property of an acute triangle Solving the functional equation $f(n^2+m)=f(n+m^2)$ Is there a different approach to evaluate $\int \ln(x)\,\mathrm{d}x?$ Operator norm is equal to max eigenvalue Is this proof that every strictly increasing $f\colon\mathbb{R\to R}$ is injective flawed? Is there a way to sum $\sum_{n=1}^{\infty}\left[ \frac{y}{n} - \tan^{-1}\left( \frac{y}{n} \right)\right]$ Why do people study algebraic extension? Confusion with regards to the phrase "exactly one of the events occurs" Find all rational points on $x^2 + y^2 = 17$ Prove the existence of graph, which doesn't contain three vertices with same degree. Show that $\sum_{k=1}^{\infty}\frac{2^{-k}}{k}=\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}$ without evaluating either sum Connectedness and path connectedness of a set whose intersection with lines is open

Recent Posts