{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

P7 - Problem 4.14 Show tho the sequence{xu deﬁned by 1 n...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Problem 4.14. Show tho: the sequence {xu} deﬁned by 1 n 15,. = (1+ ‘1'?) Solution. From the binomial expansion: xn=u+§r=g<ne=x+ime i=1 II nlﬂ—Uu-{n—JH) 1 1 + Z: n ' 3r I=l 1+;%(1“l)(1-—%).._(1_%) S1+;‘lIT(I—ﬁ)(l—ﬁ)n.(1_ill) = (1+ﬁ'l'YH-I =In+1. ['5‘ a convergent sequence. E! 3 Thus. 1:,l T holds. Also, note that for n 2 2 we have " n n x” =1+Zﬁ(1*%)'“(1-'€,—‘) 52+Zﬁ52+2§i= 53. i=1 i=2 i=2 By 'I'hooxem 4.3, [Ia] converges. (Of course, limx" = e = 2.718 - . - .) Problem 4.15. Assume that a sequence {x,,} sarigles lxn+l — J:ml S ﬂllxﬂ ‘xn—ll for n = 2, 3. and someﬁxed 0 < a < 1. Show that {xx} 1': a convergent sequence. Solution. Let c = Ix; — x. |. An easy inductive argument shows that for each n we have lx,,+1 — x,.| 5 cat""'. Thus, P — |1n+p — xnl 5 23an _xn+£—ll ‘5 Cilia-H—z S TECH-1 l=l n'IIl holds for all n and all p. Since limo" = 0, it follows that [x,,] is a Cauchy sequence, and hence. a convergent sequence. Problem 4.16. Show that the sequence [1.]. deﬁned by 1 3+x.. .n=1andx,.+1= for n=1.2,.... converge: and determine its limit. Solution. Clearly. .t. > 0 holds for each n. Now. note that - — _L.. .. _1. l-Tn+l ‘v‘nl — 3+.“ 3+1“I _ Ix —.r.7 | _L _ — (mm +§..-o 5 91‘" "HI holds for n = 2. 3. . . . .13)! Problem 4. 15. the sequence Ix”! converges If lim 1,. :1; then x 2 0 and l 1 "=1‘"'*»+'=m=3+.- Solving the equation, we get x = —_3+2£_ ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online