Question

If power series has radius of convergence r, then radius of convergence of an z^ 2n

Answer 1

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8down voteaccepted

(i) Let un=z3n2nun=z3n2n so

|un+1un|=2n|z|32n+2→|z|3<1⟺|z|<1|un+1un|=2n|z|32n+2→|z|3<1⟺|z|<1

so

R=1R=1

(ii) Let vn=(2n+i)(z−i)nvn=(2n+i)(z−i)n so

|vn+1vn|=|(z−i)(2n+1+i)2n+i|→2|z−i|<1⟺z∈D(i,12)|vn+1vn|=|(z−i)(2n+1+i)2n+i|→2|z−i|<1⟺z∈D(i,12)

so

R=12R=12

(iii) Let wn=zn!wn=zn! so

|wn+1wn|→ℓ=0⟺|z|<1|wn+1wn|→ℓ=0⟺|z|<1

so

R=1




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