Find the radius of convergence of the power

Rubi Riggs

Rubi Riggs

Answered question

2022-04-05

Find the radius of convergence of the power series:
n01(n+3)2zn
Describes the convergence domain (Ω) and study the convergence point on the border (Ω).

Answer & Explanation

carlosegundoacyg

carlosegundoacyg

Beginner2022-04-06Added 10 answers

Since
limn|zn+1(n+4)2(n+3)2zn|=limn|n+3n+4|2|z|=|z|,
therefore if |z|<1, then the series converges. For |z|=1 the series is absolutely convergent (and therefore convergent) because
n=0|z|n(n+3)2=n=31n2=π26114
Hence the series converges in Ω={zC: |z|1}

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