Question

3) Let F(x) = {* In In(1+t) dt. t (a) Find the Maclaurin series for F: (b) Use the series in part (a) to evaluate F(-1) exactly and use the result to state its interval of convergence. (c) Approximate F(1) to three decimals. (Hint: Look for an alternating series. )

Answer 1

X 3) F(x) = m(ltt) at t 0 00 m+ HA met Devding with 't 12 2 + 13 3 +4 4 m mn use know that miitt) = ? =t-t2 + 2 we get hitt) + Ct t |-} + - + tat Substituting nett) in weget F(x) = 5 (- + + + + + + 3 0.2 dt- t" x t ) dt xt2 + 3 0

= [+]"-1 + 1 - 30+ Sendt = เกษ n. 42 4. x no 75 Sfaxudx=FG)F6 + 16 Tin F(x) = x - x² 24 16+ ol -za n=1 4 b) Substituting in F(x) we get 25-1, F(-1) = a sight n=1 m2 ant n2 m n m+n 2 X=X E (1) - nal 27 m2 1 m2 6-10 = 1. E n=) 00 1 F1+1) = - S- -t, an n- 1 dn [thili som F(-1) = 1

By alternating series test E-12 is absolutely convenge- FH) = 1) converges for all my e) F (Z) = t - 111 + 2 + + (+13 = - to + 65 44 0.4.513 We know that (R₂) <100) (hugenenal Rnl <lantil) where as is the 4th term in the alternating Series