Question

Show all of your work! 1. Find a power series representation of the function f(0) = x arctan(2.c) and determine the raidus of convergence.

Please solve all.

Answer 1

So iriven that flas x arctám (2x) For as tem (22) Caretan (2x)) = 2 Now Ltx 1t 4x² - (-1). xn. 620 Replace it by 4x² i +402 Now 142 = 28 con Cascajn. »j a rclan ax = s barzdx = 28 JC-18" (4x2jn da nzo

= 2 $ $6-15 "an. scandal in an acan do M8 om anti 25 (-on on a tci - anti (un genti x 2h+1 C 6 Hence froll = oL aretan (204 C-iga. 22ntl 22+1 no anti calon caion 26+2 2 20tl x anti to ny that e know - .. • areienx = x-e een hoge Ŝ c-ijk.2x?st! & are danx = x (x-repeat an nzo o ) Šeim -32n+? anti neo

the radius of How to determine convergence. Here. qente e come Intl ant 2 Int2 Ign+q 042 here to van de as the car , Jam Line (2n+1)2nte Ř e to leantel Smite น่ช) nto =) Now = 1 D) Rat » RO.C.1. criven flos - Ni+X Find maclawin s (a) [de the azo)

flow = Sita we know that by Maclawrin f(x) = f(o) + f cojic + f(0) x 2 + series 21 * + f(o) 234. 3b no 9 sati Now proc) Sitx - flo) = sito ='(2) : - 5) (+) - 4 f" co textusble => f"(0) = F"(x) = sange 33f"60) f" (2) = - 15 ) fu (0) = -1 1 1 42 16 (6+1)/2 105 f (a) = 32(x+199423) fUCOJlos 32 we get

-160 = 1 + 2 x – 1 3 2 ² + 1 - 5 2 x Eittxt § 3-13**' 1-3-5.7.-(2-3) ne on.nl we know that (Hajk = 1+ka & K CK-1) 2 2 - x² + K (K-1) (K-2/3 x CKD (4-2).. -- + K (K-1) (K-2) ---(k-hts t ** tine integer o series will converges for st all & from – ao to top series converges on (-1) • If k2on on [-1, 1] & If I LK Loo (-1,1]

fu afitx = (1+aste Here k = I 7,0 =) series convergery on [1,1] Hence Radius of convergence = 1. o contre co f centred (b) find the taylor series of at 9=3 S013 fox) = sier ») f(3) = Ja = ple) i e vogntog f'(3) . - *"000) = 4 (2+1)*123) f"= :14) 1? = 92.253 f"(x) = 3 8( x +1) 512=) p'"( 3) - 8 (4)512 - 9x 95 f (0) = -15 16 (2+1) 7/2 = f(3) = -15 94.27 = -15 P(x) = 105 ) f (g) - 105 los 320x+1) 812

f(se) = f(a) + f'(a) f(-a)' + F"ca) (x-aj? 15 21 flu ca) Cx-a)3 + --- - 36 d at az 3 we get 40208 et f(x) = 2 + . le ( x 32 92 4 tunc