✔ 最佳答案
Qa.
a^x= e^(x lna)=1+ x lna + (x lna)^2 /2!+...+(x lna)^n /n!+...
2^(-x^2)= e^(-x^2 ln2)=1- x^2 ln2+ (x^2 ln2)^2 /2! - ...+(-1)^n (x^2 ln2)^n /n!+...
f(x)=x*2^(-x^2)= x- x^3 ln2 + x^5 (ln2)^2/ 2!+...+(-1)^n x^(2n+1) *(ln2)^n /n!+...
f'(x)=1- 3ln2 x^2 + 5(ln2)^2 x^4/2!+...+(-1)^n *(2n+1) (ln2)^n x^(2n) /n!+...
for x in R.
Qb.
1/(1+t^2)= 1-t^2+t^4-t^6+...+(-1)^n t^(2n)+...
Integrate both with respect to t from t=0~x, then
arctan(x)= x- x^3 /3 + x^5/ 5+...+(-1)^n x^(2n+1) /(2n+1)+...
f(x)=x*arctan(x)= x^2 - x^4/ 3+ x^6/ 5+...+(-1)^n x^(2n+2) /(2n+1)+...
f'(x)= 2x -(4/3) x^3 + (6/5) x^5+...+(-1)^n (2n+2)/(2n+1) x^(2n+1) +....
for -1<= x <= 1