find the exact interval of convergence of the below: in=0·nConsider 2-³n. This is deceptive, and it is tempting to cal-culate 3 = lim sup(2-2)1/n = and conclude R = 2. This is wrongbecause 2 is the coefficient of x³n not xn, and the calculation ofB must involve the coefficients an of x^. We need to handle thisseries more carefully. The series can be written 0 anx" where= 2-k and an == 0 if n is not a multiple of 3. We calculate Bby using the subsequence of all nonzero terms, i.e., the subsequencegiven by a(k)= 3k. This yieldsазк=ß = lim sup |an|¹/n=lim |a3k|¹k→∞1/3k=∞lim (2-k)1/3k 2-1/3k→∞Therefore the radius of convergence is R=1/² = 2¹/3One may consider more general power series of the formΣan(x - xo)",n=0=U(*)where xo is a fixed real number, but they reduce to series of theform o any by the change of variable y = x − xo. The intervalof convergence for the series (*) will be an interval centered at xo.2=0

Answered: Image /qna-images/answer/f048a73a-f270-44d9-bdb8-a35c5c8b5778.jpg

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

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

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter9: Sequences, Probability And Counting Theory

Section9.4: Series And Their Notations

Problem 10TI: Determine whether the sum of the infinite series is defined. 24+(12)+6+(3)+

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