Question

4. Let f(x)= (x-4)" - (n)(n+1) a) Find the interval of convergence of f. b) Find f'(x), and determine its interval of convergence,

Answer 1

a) n20 and nyo 4) We know that if a Power series { an (ox-xon be such that anto for all a lim I am an I, then plea an cau xo)" Converges absolutely for all real a satisfying (x-ko) FR Given that, 1 (0-4) » T: (711) Comparing given Power series to the Power series of an (x=%)" we get fion) m20 20 = 4 ana nini X Now, a lim 1 mento) nyo (n+1) (912) lim n(n+1) nyo (nal) (742) lim (9) nya (14th) 11+30) I Am (1) (1h) (14%) no lim nara (1440) (1+0) (140) 1 RG 1 :: The given power series converges absolutely for all real a Satisfying 19-4121 ay a 2 aru el a 32025

Now check at the end points when ar 3 The given series is & (3-4) nzo n (0+1) con não n(n+1) Ě Uneljn, where Una. n1941) ก70 Now, Untl Un (1H1) (942) nenti) nt) unl 2 un in+1) [ 2 - 1 xin -n-2 (42) na -2 n1n11) (2) ay unti-unco & NEN ay & NEM ay {un] is a monotonic decreasing sequence of positive real numbers, A180, lim nyoo o test the series & nint)) is convergent when a = 5 E The given series is n (1) lim una nimi) nya (-)" nco By Leibnitz's (514) 9 nco olang nimi no un, where niny una no

1 (x-xo)n-1 obtained by term-by- Now, Uma no nco ） nim nha vnen we know that the series Ehpis Convergent if pal th2 is convergent 4. By comparison test, the series una nini) is convergent. & The interval of convergence of the given Power 'Series Given that fra) = 8 (2-4) (111) qy fim) = (x-4) (x-4) + (9-4)2 (9-4)2 + farley (4-4)3 + (0-4) n20 neo b ) ។ 4.5 2.3 12 3:4 How differenciale (**) with respect to an 1' = the (9-4) 2 o cana)? 3 4 5 E (acyjny no nan り nal TMA) we know that, if R be the sadius of Convergence of the power series & an x-x)" Then the radius of Convergence of the Power Series nay term differentiation, is also R. :. The radius of converges of fla) is also 1 .. The power series converges absolutel for all real a absolutely sotisfying

19-421 a ma 421 a 32 n 25 Ź (3-4) & when a B, Then the series is no nt) con (11) nal nal :: By Leibnitz's test the series els convergent: when no5 Then the series (1) is Σ [ 2 041 nal series. This is an divergent :. The internal of Convergence of f'im) is [3,5)