请问这道题怎么求幂级数的系数和收敛半径？

Question

如题所述

Answer 1

f(x)=2/(1+25x^2) c0=2
f'(x)=-100x/(1+25x^2)^2 c1=0
f''(x)=-100/(1+25x^2)^2+10000x^2/(1+25x^2)^3 c2=-50
f'''(x)=-10000x/(1+25x^2)^3+20000x/(1+25x^2)^3-500000x^3/(1+25x^2)^4

=10000x/(1+25x^2)^3-500000x^3/(1+25x^2)^4 c3=0
f(4)(x)=10000/(1+25x^2)^3-500000x/(1+25x^2)^4-1500000x^2/(1+25x^2)^4 [phone]x^4/(1+25x^2)^5
c4=10000/4!
∴ c0=2 c1=0 c2=-50 c3=0 c4=10000/4!