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P s ¬ ð Ã Ê p s ¬ ð!. Theorem If f(x)= ¥ å n=0 c nx n has a radius of convergence R, then 1 f0(x) has the same radius of convergence and f0(x)= å n=1 c nnx n1 = ¥ å n=0 c n1(n1)x 2 Z f(x)dx has the same radius of convergence and Z f(x)dx =C ¥ å n=0 c n n1 xn1 NOTE When differentiating a power series, the starting index will change if the ﬁrst term is constant. E ¥ Ö Ú È ð Ã 1 3 ¥ ¢ ¦ ¼ Å Ã ì Ï Ã Õ È Ó Ë ì Ã Î Ë Ï Ã Ê Ã Ù Ë ï Ç Ã Õ È Ò Ñ Ç Ô Õ Ã Í Ð È Ç Ã Ó Ã Ê Ï Ë Û ï Ã Û Ñ Ð Ã Ú Ë Ð Ö Ð Ã Í Ñ ì Ë Ö Ú Ë Û, Ç Ã Õ Ë.

å n=1 cne ¡k2 nt sinknx, (436) Imposing the boundary conditions (428) gives u(0,t) = 2x ¡ x2 = ¥ å n=1 cn sinknx, (437) It is important to know that the cn’s are not given by the formula cn = 2 2 Z 2 0 (2x ¡ x2)sinknx,dx as usual The reason for this is that the kn’s are not equally spaced So it is necessary to examine (437) on. Fþf÷h g fê'¨ í fÜfÒg g v ¼fçfúfÔ&ì Øfû a5 fþ 5 fçfö q#Ýfég f¹ ¾h '¨ §"'h. Chap XÏne If«²li¡ Š¢sam– , ¤ ¡ eBookœ!e ÏøWõ ˜ÜÐ( DƒGa—RÝ€m^úÄ ®u´~òé ñe 9§\ ‘8æáñÙ’Ø˜ è l¶õ >¾MÌç¿ þ ysë”ß• Î˜Z 8 O QÂäu ° 4• ÁëL¦èez«^µµ›ÏSZbkæ¹J•XA )Ü 0d ª#4ü Ô 1ÜÐ¸5KÖn ,—Y2ÜGÝ Ùì;ÃC}í_A—bØ>L¢ ¶ Ÿ ^ )Oõ ðì¢°L.

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