Question

Find the intervals of convergence of f(x), f'(x), f"(x), and the indefinte integral of f(x) dx. (Be sure to include a check for convergence at the endpoints if the intervals. Enter your answer using interva notation.)

Find the intervals of convergence of f(x), f'(x), F"(x), and Sf(x) dx. (Be sure to include a check for convergene Ax) = Ë (-1)" + 2(x – 7)" n=1 מלח (a) f(x) (b) f(x) (c) F"(x) (d) f(x) dx

Answer 1

Given that f(a)= Ï cuta" (x-7)" non n=1 cliff wort.nl f'(n) =E Guynes ni (4-7)nal n7" Eight (2-7)-1 f'(a) M8 74 nal Again cliff w.e.t.ly' ཤརྣ ) ร (M-1) (21-734-2 7 n=1 Integrate equ (1-7) ela { fonde = 1 nal nzn (x-7)+1 n=1 n(n+1) 7 Now, convergence of f(n): from f(n) = E Light (2-7) nan nal here this power series will concurge for n=7 we'll use the ratio test. L=h anal Line Eught? (~n7]44 (n+1) qual yo cvm carcasa Lin (2-2) | = 12-71 an imtina (n+1) 7 > #12-71

L=1 Ratio test tells us that L<1 the series will converge, if L>l the series will diverge and if ull don't know that what will happen. 11-7141 => 131-71<7 series converges – 12-71 > =) 17-71 >7 series Diverges The Raduis of convergence for power series=7. Now Interval of convergence -7 < x-7 27 Ocak 14 So, most of the interval of validity is given by okr< 14 All ull need to do is determine if the power series will converge or diverse at the end points. At n=0 È tutt'<3) A n=1 n7 (༤༢༨ ། n79 nal th nal a This is harmonie series and ut is diverges At n=14 c ntl 71 n nu nal Eight nzu This is alternating harmonie series and it is converges The Internal of convergence oxx414 - So, 14

Convergence of f(n) : f(n) = 8 (night (n-3)n-1 an n=1 Ratio test 7 La a time ku ta 2 (n-3) 7+1 (wynt (4-7)n-1 (N-7) 7 712-71 ↑ 12-7141 = 171-7127 series converges $12-71>1 => 17127137 series Diverses R.O.C. for power series=7 Interval of convergence ocne 14 At n=o (-12h 7n ght €7) - 을 놀 This is Divergent. uc At n=14 į sapning = does not exist M8 7 nal n=1 구리 nal Ě right ani 74 nal This is Divergent Interval of convergence OLNL 14 -> (14)

convergence of f(n):- P f"(a)= Ě (-1 (1-1) (2-7)-2 74 Ratio Test La respond in con-a) *** _ 74 (ight (1) 61-7) 4-2 fin n(n-7) 7 (na) = 17-71. I 7 10-71 <1 12-7127 10-7171 10-2177 converges Diverses R.O.C. of Power series =7 Interval of convergence 0<x< 14 At n=0 2n-1 Ź caughty (n-1) Ca) 1-2 GB nel 74 nal 72 Divergent P Does not enist - nal At na 14 Ž Light (n.1) 742 Gynt (mal) nal 74 72 Divergent. Internal of convergence olna 14 (14)

tc convergence of Stonian : S$(*)dx = Ë coohal Can-a) mai in (n+1) an Ratio Test La verso lana en historien (mam n(n+1) 7" (n+1)(n+2) 7h41" Eight (1-7)+1 Lin -n (n-7) M+2) 7 - 17 12711 - 12-7147 converges 12-7171 =) 17-7177 Diverges R.O.C. for power seris = 7 Intereal 0<x< 14 At no 을 (night! (-7)+1 n(n+1) 7 n(n+1) 7" (L3nt2 7 n=1 7 n(n+1) convergent. At x = 14 . MBP (rignal 7+1 Z c-ynay n(n+1) Well nel n(n+1) 74 convergent Interval of convergence OSN314 [o, 14