Question

1. Represent the function 10/1−10x as a power series f(x)=∞∑n=0cn x^n
Compute the first few coefficients of this power series:
c0=   
c1=   
c2=   
c3=   
c4=
Find the radius of convergence R=

2. The Taylor series for f(x)=e^x at a = 2 is ∞∑n=0 cn(x−2)^n.
Find the first few coefficients.
c0=   
c1=   
c2=
c3=
c4=

Answer 1

@ We serits for know the I 1-x no power an Š (10x)" (qla xay for'} I-102 no Ž 10n.am I-lox No multiply both sides by io' & 8 WAS SW lonti To 1-102 an ; (i 10h 10's lonti] -> 10 1-lox E Cna" where no Cn = n+1 10 lol :: Co Ary 10 С. 102 = cetry lovo C, 103 70 3+1 C3 = (n+1)+1 Now, Cuti lonta The radius of convergence is : R lt lonti lt nyo lt Cn Cuti You) no lonta nyoo R - Ans lo

=ea dx x=2 We have, f(x) = ex f (x) = f'(x) = f'(x) = f(x) = en {: d (ex Thus, f (2) = f'(2) = f (2) = f'(al = f (2) - e² power series of f(x) about is given as : f(x) = f(2) + f'(2)(x-2) + F"(2). (1-2) + f'"(2) (x-2)3 31 f(x) = e² + 0² (x-2) + e² (x-2)2 + eZ (x-2)3+ 2! 3! f(0) = 3 62 (7-2) Ś Cn (x-2) The 21 я - neo n=o : (n = e² n! co= 2 Z e 13 e (10!-!!! =) 2 ci 11 C2 = 2 e 21 2م 11 (2x1) 2 2 C3 e 31 6 2 (3x2x1) e² 4X34281 Cu - 4! 24 Now, Cnti e² (n+1)! (n+i)ni Thus, Cn (n+1). ni = lt (n+1) R = lt lt no e2 n! х nto nyo Cuti 02 - o cong