This week, we’ll work out some Taylor Series expansions of popular functions.
When x = 0
\(1+x+x^2+x^3+x^4+x^5+O\left(x^6\right)\)
\(\frac{1}{1-x}=\sum _{n=0}^{\infty } x^n\text{/;}\left| x\right| <1\)
When x = 0
\(1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\frac{x^5}{120}+O\left(x^6\right)\)
\(e^x=\sum _{k=0}^{\infty } \frac{x^k}{k!}\)
When x = 0
\(x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\frac{x^5}{5}-\frac{x^6}{6}+O\left(x^7\right)\)
\(\log (x+1)=-\sum _{k=1}^{\infty } \frac{(-1)^k x^k}{k}\text{/;}\left| x\right| <1\)