Question

4. Derive the Maclaurin series for In (1 + x) 5. Derive the Maclaurin series for In (1 – x) 6. Derive the Maclaurin series for In (*) 7. Find the Maclaurin series for x sin x (1-x)

Answer 1

Given that 4) Derive the mailauris series 557 ln (1+1) we know the general expression for the Mayaurins fa) = f(+8'609x+f"Cova --+f"(0) gäven that f®) – en (1+x) - © using x=0, the given equation o become f®)= en 1+o) - Inico . Now-faking the derivatives of the given furition and using x=0, we have $$'@= its = (1+27" s'o =(1+017-1 => $"62 - Potwia , f'(o) - video y ->$"(29 t"") = (tot } = 2 = f'(o) - Ontem, CO 06034 = -6 Now cusing maclawin' Series expremion faction, we have f@=-50+$'(o) x + a $"694, 3063*for -- en (1+) = 04x0) + 6 197*(23+* (-6+ --- in (ita) = x - 2) + Gof-. en (1+x)= x-ramp - - Mackurin series for en (Ita) is en (+) = xwe û t--

5) derive maclaurin serice bol In (1-») consider the funefeon (16) = ln (1-4)) - © wing x=0, the g equatém @ becomes 5663 = en (1-0) = n1 =0. Now, take the derivatives of the given fungeon and wing x=0, we have = f'@= jk , sle) = 10,7 = -1 *tag - Photogr , sono inox - 7-4 Öt"(n = fºo) = 2-7--2- *$a = a aby ste) = M = - = -6 Now, using the genery form of Machuris series, we have FØD= 40+ floga te t'ont 304** *0 ln (1-1) 0 +*813 + * 641)+ (-1)+ + 50+ -- 10(1-1) = 0-2 + CD + * (-277 Cofno In(-1) = - Therefore gaclurin series of souls (mn) is. | | (_x) = -4-- -7

by Derive the marfaurin series for en (177) consider ttu funition fo) = (443 : We know the Macfawrin sering tot in (+1) and esca 1) (+)= x--e. en (-1) =-* Using the property of logs we can write the sligione funzion as in (4+) = ln (1+) –- ls (1-1) Then substitutions the above two series we get AD (4+)-[a-*-7-{-*- en (112) = 24+249 2** f. -~- Therefon Maclauris series for en (141) is Tln (4+*2 = 24+2 +2

Ty find the Maclaurin Series 687 a sine Con Side ftu su- ) == &p We find the maclaurin series for a sina, forsty Derive Hu Maclawin expansion 679 sina aut Multiply it sya.. So let's derive the Macfaurin expansion for sent let g(n) = sinx : Them 36 = ) " general Expansion) 0-0 0 .^ 961) = sina ; 90 = sin(O) = 0 86 - Cosa, g'(0) = COSCO) = 1. La'6) = – sản* ,9*2 = -Sử () = 0 . 3 %*) --Cosa, 9" (0) = -coaf Then (4k+n) Hance, sona el a ::: : **-- single = 12nt, Glj?2011 46 = sina = a E l não canti) esinx = fm = & A" Therefore, the Maclauris sering to a sihe is Casinx = 8 Gj" 21+2 720 (anti) (n) = o Con - x = swi no (29+1)! n20 (2n+!