Find a power series representation and What is the radius of convergence, R for all

Question

a) f(x)= ln(1+x)

(b) Use part (a) to find a power series for f(x) = x ln(1 + x).

(c) Use part (a) to find a power series for f(x) = ln(1 + x⁴).

Answer 1

a). ln(1+x) = ∑_{(n=0 to ∞)} (-1)ⁿxⁿ⁺¹/ (n+1)

To determine R, the radius of convergence, use the Ratio Test.

l a_n+1 / a_n l = l (xⁿ⁺²/ (n+2))((n+1)/ xⁿ⁺¹) l = l x l [(n + 1) / (n + 2)] = l x l [(1 + 1/n) / (1 + 2/n)]

lim_{n→ ∞} l a_n+1 / a_n l = l x l. By the Ratio Test, the series converges if l x l < 1. That is, -1 < x < 1. The

series also converges at 1, but not at -1. R = radius of convergence = (1 - (-1))/2 = 1.

R = 1 for parts b and c as well (use the Ratio Test).

Answer 2

a)

f '(x)=1/(1+x) = ∑_n=0^∞(-1)ⁿxⁿ

Integrate

f(x) =∑(-1)ⁿxⁿ⁺¹/(n+1)

b) xln(1+x)=x∑(-1)ⁿxⁿ⁺¹/(n+1)=∑(-1)ⁿxⁿ⁺²/(n+1)

c)

ln(1+x⁴)=∑(-1)ⁿ(x⁴)ⁿ⁺¹/(n+1)

All series diverge