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Unformatted text preview: . Thus the interval of convergence is ( 1 / 3 , 1 / 3). 23.6b. Give an example of a series whose interval of convergence is exactly ( 1 , 1]. Solution. The series n> ( x ) n /n converges (to ln(1+ x )) when  x  < 1 (by the root test), diverges at x = 1 (since 1 /n = , and converges at x = +1 as an alternating series whose terms 1 /n monotonically tend to 0 in the absolute value. 23.8. Show that f n ( x ) := n 1 sin nx are dierentiable, tend to 0 for all x R , but lim f n ( x ) need not exist (at x = for instance). Solution. Indeed, f n ( x ) tends to 0 since  n 1 sin nx  n = 1 /n as n . However the sequence f n ( x ) = cos nx turns into ( 1) n at x = which has no limit as n . 1...
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 Spring '09
 Taylor Series

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