Question

(13) For what values of x, does the power series 1 n. (2x + 1)^ In 5n n=0 converge? (14) For what values of x, does the power series ^ (n + 1)! nao (2n + 1).X" converge?

Answer 1

0.13. i to (2x+1) no n. 5 which states that Since we know Cauchy's Root Test Hf Eun is a sisies of positive tesms luch that (a) limit (un) ' I then (i) Eun is convergent if ki (n) Sun is divergent if e> Eun may converge os diverge . n- fails if e=1 (or we for we at l = 15e say test Also lt (2) - nao from question : Un= 1 (2x+1)". 1. n. 5n met, console in one C 2271) from @ I CQxH) .. lt ..? nooi) 5"? . Here e = 2x+1 5 s

5 so if e ki then sun is convergent which means if axt! <l then series convergent So 2x + < -> ax+ <5. a) 2x <4 = [224] 1. Thus for [x<4 Given series converges 114. 8 (n+1)! 2 nzo (2n+1)! . : since we know . D'Alembert's Ratio Test - which states of sun is series of positive Terms such that ras dinit un = 1 then (i) Eun is convergent if l>i o (ii) Ein is divergent if eci 10 l) E Un may connerge oi diverge higher at lie test fáble if e=1) T (6) If et un = then Eun il convergent D © 1-0 unt

Now from question =10+1) + +1) +1)! Unt ( ( n+1+1)! (@n+1)! = (nta (2n+3) u using Ratio Test (n+1)! on lt n un unti (2n +1)! (n+2)! ant (2n+3)! nn) = lt (n+1)! (2n+3)! x n30 20+1)! (^+2)! dince = n(n-1)(–2).... 3.2.1) 3 et 1(+1)! (20+3)(2n+2)(am+)! no alent)? (n+a) (n+1) = net (21+3)(2n+2) (n+2)x Filt (2n+3)(2)(n+1) (n+2) x It (2)(2+3)(1+4).m2 7400 120 () (n)(1+2

= lt (2) n (2+3)(1+-) no (1+) o where at Snice at x = nyo n (2+0)(1+0) = lt no (2 (1+0) et (12)(20) Thus If 4 >I then Eun is convergent . 4x i for [n<4] linen series is convergent.