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8.7 (odd) - SECTION TAYLOR MACLAURIN 8.7 AND SERIEStr 253...

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SECTION 8.7 TAYLOR AND MACLAURIN SERIES tr 253 39, By Example 7, tan-1, : i (-1)' #for lrl < 1. In particular, ro, " -- ft, . ir , /+) (_r),!44)"*' ,, )'+=1. wehave - ta"-'(r7al i!o' znT!-: t (-t)" (.i/ ,/i2n+L' 6 @ (-1)" -& r-1)' sor:ft Affi-2v3EodiF 8,7 Taylor and Maclaurin Series 1, Using Theorem 5 with fl bn(n s)n,bn ry,so b3 ry 3, Since 7t"r(0) : (n l1)!, Equation 7 gives the Maclaurin series /t?lo)"": (rz+t)t"': (ntr)r^.ApplyingtheRatioTestwith an:(n-tt)r^givesus n=O IL. n:O tLi n:O Iim l4l: u* lg:# l:1"1 t,* ?I?:lol.r:lrl.Forconversence,wemusthavelzl <1,sothe n-*l o,n I n-ol (n+I)r" | "n-*n*I radius of convereence .R 1. We use Equation 7 with /(r) cos 0. cosr /(0) + !'(o)r + f"!0) ,'. qP r' * /(n].(o) xn +... ^2 -4 {: (_t). ur. -1 _* :L_ 2! 4l nu:o Qn)l ( -1\n r2n rf an : vffi,th", n yr*t (r) ;t"r (0) 0 I 2 3 4 cos 0 -sinr -cos0 sinr cos c 1 0 -1 0 1 I ,. la,*t L. I rzn+z Qdll , "'* l;l :,r* lW+fi.'-61: '- :0<L forallo. So R: m (Ratio Test). or,: S [email protected])(o). .: S 5'., nl * ilo nl* lo,*t | ,n+r Wf+l nt I n+6 an .*- L \r, + I,ll o'' J :0 < 1 forallr So ft: oo. t,- 1 i'JJ (2n + 2) (2n + I) blul :llm- i-*n*l n 7t-)@) 0 I 2 3 A ^5o e 5e5' 72 ^5r r3 ^5o 5ae5' : 1 o 25 125 625
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254 U CHAPTERS SERIES I o ifn is even e2n+r f(n'f0): { sosinh.r: )- ," I t lf n is odd Fo (2n + r)l' Use the Ratio Test to find R. If a,,. :;",*1, tn"n \zn + r)! ,. o,+, | ,2rtr3 + l)l , t ";Ll o,, ,,:;l(2n-J)l r.2n-r - i-'*(2n+J)(2tt+2) :0<lforallr So r? : oo. )^60 f (") :7 r 5(r 2) + [email protected] *')' * ,Dr:(' 2)" :7-t5(u-2)+(r-2)2 Since n. : 0 for large n, R m. cos., :^i,{P,'-,r)k : -,*(' tin' -"#* (' ui") -. . 6 I- *\2n _ \- / 11nfl\r -/'/ ,?o' (2n)l 't7, n \")(r) yr"r (0) 0 1 z i sinh z cosh a sinh r cosh r 0 1 0 1 0 n yt"t (r) yr"t(2) 0 1 J ,4 Ilrl:r2 l-t2r z 0 0 ; 7 z 0 0 13. clearly, [email protected]): €-, so yr")(3) e3 and,r' "la B)". If a", fiO* 3)', then rm lgrtll: t,- l"'('-r)"+' = "' ,=l: ri,r, k+:0< r rbrarrr,solr:oo. "*tl ,;-;l (rr+l)! c3(,r-3)"1 ';-; n+l ii n- 11. 19. rf If, lr t I' lzl reF 23. Th (1 *, So 15.
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8.7 (odd) - SECTION TAYLOR MACLAURIN 8.7 AND SERIEStr 253...

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