Question

Find a power series representation centered around the zero for the function, and give the convergence interval.

a)  

b)  

c)  

d)  

4 2 3 (5 2x) 1 2r 12 xln (1

Answer 1

Date 4ta2 Reorite this Reminds 1æli provided 1-X 20 Here て1) 2. ムナx2 Interval ot convergence is 2n 2 2 T 2 el2

15+2x)5 5(1+ 215 )3 let 1 +2152) lt2151)+(-2152)+2152) L.HS 2/5 d2 1-(-2152) R.H.S C1-1-215x)) 우+(-215x)+f-215 22+ (-215 x)와. 2 +2- again derivaive at H.s&R.HS 4/25 C1--2152)3 215 2 R.H.S 3 4 +4.3 + 3.2/-2 - 2 5 4125 (L-(-2152))3 3. 2 - 2 1e. 4.3 1 (-1) nin-1a2 n 2 -2 C1ncn-215x) CD In ter val ot convergence -1 lie iintD 15 0-2 +252 ten -1 Y)(n)(-152

Pege No. Dato 1-x 1+21 1 1-(-21) i-(-2x) (-2x) nEO 1十2X Tnterval at convergence 12x11 122l1 ll 12 2

d Nate that 12 pawer series oh 1-(-1) on both sides wrt-x abegicaling we 1t 020 n smittarty In (14x) こン 20t 2 C-1) n+ Etuo n=0 ind the intercal at conrergence ntt 20+5 C-12 Lelim nつの (120Cnt2) im nうの 2n+2