Consider n=0 2-³n. This is deceptive, and it is tempting to cal- culate B lim sup(2-¹)¹/n = ½ and conclude R 2. This is wrong = n=0 because 2 is the coefficient of x³n not xn, and the calculation of ß must involve the coefficients an of xn. We need to handle this series more carefully. The series can be written Σão anxª where azk = = 0 if n is not a multiple of 3. We calculate by using the subsequence of all nonzero terms, i.e., the subsequence given by σ(k) = 3k. This yields 2-k and an = B = lim sup |an|¹/n = lim |a3k|¹/3k Therefore the radius of convergence is R = }} = 2¹/³. ∞ = = _lim (2-k)¹/³k = 2−1/3 k→∞ One may consider more general power series of the form Σan(x-xo)", n=0 (*) where to is a fixed real number but they reduce to series of the

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