Question

Find the radius of convergence and the interval of convergence. Š (4x - 5) 2+1 3/2 n Find the Taylor series for the given function centered at the indicated value of a. f(x)=, 0 = 1

Answer 1

nole Ratio test: Series Ean We that have Suffose Define La lim antl no Then the LLI if con vergent is absolutely series Elence convergent series is divergent Rafio lest fails 2 i L>l, the 3. if 1=1

Given Series Pwi { Cux-524) nh ح ratio test wsing la lim antl no an 26+3 32 =) lalim la lim | (un-s) 1-)0 (n+18h (un-S) 2ne1 Le lim = n-) D (10-57* ( < 4+ 13 ) ] L= (un-st. (170) = (4.-57 = ) Now for convergence of Given series By Ratio test: (un-5) < 1 lun-sl - - R U-I и R so, Radius of convergence =

Convergence for In-5 ) и u 1 ca.5 LA u u lc a<3 2 at end points of Interval le (i et lel ratio test Now fails) Jordan un-s=1 - E (120+1 { n-1 nh n--| n=1 20+1 1 ow n = 12 - un-5=-) (P = 2 > 1) converges) by TP-sesies test (-) E nh I 1 converges in-1 nal So Interval of Convergence! '스스를 Cors sx + ['z series expansion nole, Taylor of fox) about x=a is f(x2= fras & Plal. (x-2) + P" (a) (x-ag? -flaxe, as 2 n! 8 W f (a) (x-a)

frasa ! -14 a =) f(al- 1 f(0 = 1 n2 2 -2 finga = f'(1) - 3 n f"(na f"(1) = 43.2 = 31 + 3.2 u (Q.3.47 -ul =) f (1)= -4? nu au lv SI tu!.5 = f(1) = 5 5 h f?(7)= (-1)". (27) f(x)= (-18 (nt Di an ") ( - ) Jo, fra)= I 12 M & M8 n-o = 4 (-1)". (n+1)! (n-1) il (-1)" (N+1). (1-1) M8