Visualization of the famous Euler identity with the help of this ingenious toy: https://www.matheretter.de/rechner/gfplot The screenshot shows the functions f(x)=cos(x) g(x)=-x²/2! +1 h(x)=x⁴/4!-x²/2! +1 i(x)=-x⁶/6!+x⁴/4!-x²/2! +1 j(x)=x⁸/8!-x⁶/6!+x⁴/4!-x²/2! +1 k(x)=-x¹⁰/10!+x⁸/8!+-x⁶/6!+x⁴/4!-x²/2! +1 l(x)=x¹²/12!-x¹⁰/10!+x⁸/8!-x⁶/6!+x⁴/4!-x²/2! +1 ...z(x)=x^2n/2n!-x^(2n-2)/(2n-2)!...+1
Visualization of the famous Euler identity with the help of this ingenious toy: https://www.matheretter.de/rechner/gfplot The screenshot shows the functions f(x)=sin(x) g(x)=x, h(x)=-x³/3!+x, i(x)=x⁵/5!-x³/3!+x j(x)=-x⁷/7!+x⁵/5!-x³/3! +x k(x)=x⁹/9!-x⁷/7!+x⁵/5!-x³/3!+x l(x)=-x¹¹/11!+x⁹/9!-x⁷/7!+x⁵/5!-x³/3!+x, k(x)=x¹³/13!-x¹¹/11!+x⁹/9!-x⁷/7!+x⁵/5!-x³/3!+x
The visual proof of the #Euleridentity e^ipi+1=0 with a #Taylorseries expansion with six or seven terms is nice too, isn't it?
e^ipi=cos(pi)+isin(pi)
=-1+0
#SharingIsCaring #sharingisthenewlearning #mathematics #maths #math #sine #cosine
I used this ingenious toy: www.matheretter.de/rechner/gfplot