Question 9.T.14: (Cauchy-Hadamard) The series ∑anx^n is absolutely convergent...
(Cauchy-Hadamard)
The series \sum{a_{n}x^{n}} is absolutely convergent if |x| < R and divergent if |x| > R.
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Applying the root test (Theorem 4.8), we obtain
\lim \sup |a_{n}x^{n}|^{1/n} = |x| \lim \sup |a_{n}|^{1/n} = \frac{|x|}{R}.
Hence the series converges absolutely when \frac{|x|}{R} < 1 and diverges when \frac{|x|}{R} > 1.
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