Question

4. Use the power series representaion f(t) = In(1 - 1) =- for -1 <<1, k=1 to find the power series representation for the following function(centered at 0). Give the interval of convergence of the new series. p(r) = 2.r" ln(1-2) 5. Find the power series representation for g centered at 0 by differentiating or integrating the power series of f(perhaps more than once). Give the interval of convergence for the resulting series. 1 using (3) 1-

Answer 1

Please comment if you need any clarification.

If you find my answer helpful please put thumbs up.thank you..

9 f(x) = ln (1-x) E Terci K Kel P(x) 6 = 2x In (1-2) 6 ik 6 ==2x E Kar k 6+k P(x) х -E 2 als as I Kol k is [=1,1 Interval of convergence 6 fox) - f(n): با ۱۱-) 2 f"C) (1-x)?

QU fox) in care 1-2 h²o a eo = { ne n- n1 n-2 f'xi & non- h=2 f"raj SO 11 n-2 962) - incnul) n (1-)3 2 ከr 2 gal ě non-in-2 ทะ - ALO gal (n+2)(n+1)x no 2 (63+3)(n+z/xht! no n-do 2 ndo lim antl lim. an (n+2) (htilah (h+3)(n+2) lim pal (n+2) Cnti) Tlak as lelki from x= -1 E (n+2)(n+12611 diverges na 0 2 (n+2)(n+i) (1) - diverges for xil ss E n=1 2 So interval of convergence is (-1,1).