Question

I would really appreciate your help with solving this whole question of my assignment due in a few hours. Willing to rate and like!! Thank you so much

Q1/ Solve the following series question a) is the following sequence increasing, decreasing, or not monotonic? Is the sequence bounded? a. an = 2n13 b) Use power series to approximate the definite integral to six decimal places. a. Jo 773 dx b. So 2 x ln(1 + x²) dx 0.3 x c) Show that the sequence defined by a = 1 anta = 3 - Is increasing and that an < 3 for all n. Deduce that the sequence is convergent and then find the limit. d) Determine whether the following geometric series converge or diverge. If they converge, what is the sum? An=1 12(0.73)n-1 e) Determine whether the series is convergent or divergent by representing it as a telescoping series. Ens ( min) 1) Find the radius of convergence and the interval of convergence of the series. 2n=1(-1)"nx" 8) Find a power series representation for the function and determine the radius of convergence, (compare to geometric series!) f(x) = f(x) = 2x+3 h) Use your table of MacLaurin Series to obtain a MacLaurin series for the following: f(x) = arctan (x2) f(x) = x? In(1 + x3) 1) Use Taylor's Inequality to estimate the accuracy of the given approximation f(x) = Tn(x) when x lies in the given interval: f(x) = x2/3, a = 1, n = 3, 0.8 <x< 1.2

Answer 1

(a) nanti 7 an <?(n+1) = 2n+3 <a(n+1) +3 -> I > I 2n+3 ahhit3 - an> anti a decreasing Now, no ano Also an <a & n> 80 Bounded (c) anti a 3-4 To show, an is increasing ie anti> an Put nal Then, 2 z 3-1 =27 ny a no True it is true Put - nak ie and assume aut 7, ar Put n=kt Then aut2 = 3 -1 aut 3-1 av = aut

- aut Thus aut27 auth foo au sauti so true & NEN 2174 Thus increasing and - 1 st 2 are anet Also, au > a * no 1 - an>1 x ns 1 ano & nya = -ancot n> 1 Also I co & n> 1 oo an to an Thus 3- 1 <3 * nii 8 an <3 since sequence is monotonically increasing and bounded above " convergent It alati = It (3-2 ) a et anti Elts ality 1 Slet it an alt 00 ntos nro nto an 100 7 l = 3 - 1 d² 30 +120 2 l = 3 7 N9-4 = 3IN5 Now 3-5 <1 : 043-ws 2

d) is de 3t us 8 12 60.733n+ 14 Now this series has ratio 0.73 and 10.73|<1 %. This converges 19 earnt a n=1 a 17-0.13 I-r 12 T-0.73 sum 124 6 - you . som 227.09 027.0 9 Now , Now Now lez <1 : converges sum seme = 662 (f) Ratio test Series ě an converges if lim No Mtol Now I'm 1 z 1+1 ทใeo Eign nan

Thus it converges if bel <1 : Radius of convergence is a Interval: tcaci Ź (-237 : lim | (x-237+1 DALI n2+1 neco (a+n²t (1-2) - 19-21 <1 8 Radius of convergence is 1 and interval: nlcn- 221 7 I<a<3