Question

Question # 2. (2 marks) Show that the ratio test fails to apply to the series, 7-n+(-1)", but that the root test does apply. Use the root test to determine if the series converges or not. n=0 Question # 3. (3 marks) Consider the power series, f(x) = į an(x + 1)". Suppose we know that f(-4), as a series, diverges, while f(2) converges. Determine the radius of convergence of the power series for f'(x). Precisely name the results we learned in Week 3 that you use, and where you are using them.

Answer 1

7-n+lim nel = 7-ntion anti= 7-n-1-(4) wing Ratio Test: lim = lim anti int-cuin zon + Gin an nto م = 7-3 even = lim 7-1-26-17 =)) in 기 i na odd. : Ratio Test fails. {

Using root-Test : ac L=im nto = lim 17-ntuinlyn - Mindez+++"yn 17-1+01= root Test Series converges. I <I ::. By

n-D f(x) = Ź an (x+1)? +(-4)= El an (-3)" diverger. neo 0 +(2) Σ αη 31 → convergen. Radius of converges = 3. lim anal-3n+1 nyoo | 21 { diverges} an2-3n lim lant R<3. lim anti 3+1 n100 an an un + 1 = 1 {converges} 1 * $ =) R23. =) um те . ant! an R=3. => for all Ixtil <3 f(x)= Ian (x+1)" converger f'(x)= Z nan(x+1)^-1;#1x+1k3 =) no lim ni mms | ***1) an+1(x+1)" n an (x+1)n-1 =) = lim =) = lindo la re 111* *) (**») ant! [x+))) => 1X!! <1=) {x+ık 3 - 1x+1) 3 an -) Radius = 3.