Question

y=ln(4+x)利用间接展开法展开成x的幂级数

Answer 1

解:
y=ln(4+x)= ln4+ln(1+x/4)
=ln4+ ∞∑n=1 (-1)(n-1←次方）/n(x/4)(n←次方）=ln4+ ∞∑n=1 (-1)(n-1←次方)/4(n←次方)n.x(n←次方)
∵x/4∈(-1,1】
∴y=ln(4+x)展开成x的幂级数为x∈(-4,4】.

Answer 2

f(x) =ln(4+x)
f'(x) = 1/(4+x)
f''(x) = -1/(4+x)^2
f^(n)(x) = (-1)^(n-1) . (n-1)!/(4+x)^n

f^(n)(0) /n! = (-1)^(n-1). (1/4)^n /n

f(x) = f(0) +[f'(0)/1!] x + [f''(0)/2!] x^2+....
= ln4 + (1/4)x - (1/4)^2 (x^2/2)+...+ (-1)^(n-1) .(1/4)^n .(x^n/n)+....+...