Since the series ∑ n = 0 ∞ ( 1 − b n ) {\displaystyle \sum _{n=0}^{\infty }(1-b_{n})} converges, we have
lim n → ∞ ( 1 − b n ) = 0 ⟺ lim n → ∞ b n = 1 {\displaystyle \lim _{n\to \infty }(1-b_{n})=0\iff \lim _{n\to \infty }b_{n}=1} .
Applying the ratio test, the series ∑ n = 0 ∞ b n x n {\displaystyle \sum _{n=0}^{\infty }b_{n}x^{n}} converges when
lim n → ∞ | b n + 1 x n + 1 b n x n | = lim n → ∞ | b n + 1 b n | | x | = | lim n → ∞ b n + 1 lim n → ∞ b n | | x | = | x | < 1 , {\displaystyle \lim _{n\to \infty }\left|{\frac {b_{n+1}x^{n+1}}{b_{n}x^{n}}}\right|=\lim _{n\to \infty }\left|{\frac {b_{n+1}}{b_{n}}}\right||x|=\left|{\frac {\lim _{n\to \infty }b_{n+1}}{\lim _{n\to \infty }b_{n}}}\right||x|=|x|<1,}
and diverges when | x | > 1 {\displaystyle |x|>1} . The last equality follows from lim n → ∞ b n = 1 ⟹ lim n → ∞ b n + 1 = 1. {\displaystyle \lim _{n\to \infty }b_{n}=1\implies \lim _{n\to \infty }b_{n+1}=1.}
Therefore, the radius of convergence of the power series ∑ n = 0 ∞ b n x n {\displaystyle \sum _{n=0}^{\infty }b_{n}x^{n}} is equal to 1 {\displaystyle \color {blue}1} .