(n=1~∞)∑(4^n*x^2n)/n!

Question

(n=1~∞)∑(4^n*x^2n)/n!

Answer 1

已知e^x=1+∑x^n/n!= 1+x/1!+x^2/2!+x^3/3!+...取f(t) = e^4t则f(t)的n阶导数为4^n·e^4t可得f(t)的无穷级数展开式为e^4t = 1+∑4^n·t^n/n!所有∑4^n·t^n/n!= e^4t-1令t=x²则有∑4^n·x^2n/n!= e^4x²-1...