Taylor Series

• Polynomials
• An instance of a Power Series
• $f(x)=\sum _{n=0}^{\mathrm{\infty }}\frac{f^{(n)}(a)}{n!}(x-a{)}^n$

Common Ones

• $\frac{1}{1-x}=\sum _{n=0}^{\mathrm{\infty }}{x}^n$
• $e^x=\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}$
• $\sin x=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n{x}^{2n+1}}{(2n+1)!}$
• cos is just differentiating
• sinh is just always positive

Derivation

• Take $T_{n,a}(x)=c_0+c_1(x-a)+c_2(x-a{)}^2+\cdots +c_n(x-a{)}^n$
• we want $T_{n,a}(a)$ and every derivative of that to agree with $f(a)$ and every derivative of that
• $T_{n,a}(a)=c_0=f(a)$
• $T_{n,a}^{\prime }(x)=c_1+2c_2(x-a)+\cdots +n{c}_n(x-a{)}^{n-1}$
• $T_{n,a}^{\prime }(a)=c_1=f^{\prime }(a)$
• $T_{n,a}^{″}(x)=2c_2+\cdots +n(n-1)c_n(x-a{)}^{n-2}$
• $T_{n,a}^{″}(a)=2c_2=f^{″}(a)$
• So in general, $c_n=\frac{f^{(n)}(a)}{n!}$

Maclaurin Series

• Just the case where $a=0$
• Only odd powers appear for odd functions, and only even powers appear for even functions, since the odd terms are odd and the even terms are even

Shortcuts

• You can find the Maclaurin series of $f(x^a)=g(x)$ by using $T_{k,0,g}(x)=T_{n,0,f}(x^a)$ (k comes from the highest exponent you get)
  • Using this, you can find $g^{(n)}(x)$, where $g(x)=f(x^n)$, by taking $g^{(2n)}(x)=\frac{(2n)!f^{(n)}(x)}{n!}$
• If you have $f(x)=T_{n,a}(x)$, then $f^{\prime }(x)=T_{n,a}^{\prime }(x)$