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Unformatted text preview: en 2 t sin nx. 17. Fix t > 0 and consider the solution at time t = t : u ( x, t ) = ∞ ± n =1 b n sin nπ L xeλ 2 n t . We will show that this series converges uniformly for all x (not just 0 ≤ x ≤ L ) by appealing to the Weierstrass Mtest. For this purpose, it suﬃces to establish the following two inequalities: (a) µ µ b n sin nπ L xeλ 2 n t µ µ ≤ M n for all x ; and (b) ∑ ∞ n =1 M n < ∞ . To establish (a), note that  b n  = µ µ µ µ 2 L ² L f ( x ) sin nπ L x dx µ µ µ µ ≤ 2 L ² L µ µ µ f ( x ) sin nπ L x µ µ µ dx (The absolute value of the integral is ≤ the integral of the absolute value.) ≤ 2 L ² L  f ( x )  dx = A (because  sin u  ≤ 1 for all u ) ....
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This note was uploaded on 12/22/2011 for the course MAP 3305 taught by Professor Stuartchalk during the Fall '11 term at UNF.
 Fall '11
 StuartChalk

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