{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# P8 - Problem 4.17 Consider the sequence{In of real numbers...

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 4.17. Consider the sequence {In} of real numbers deﬁned by x1 = l and x,.+1 = l + 1—17 for n = 1.2.... . Show that {In} is a convergenr sequence and that lim 1,. = ﬂ. Solution. _l"l1'1ea.'~=3.~inchilctiveargunumtxshowsthaws,l > Oforeech n. This implies that, in fact, we have 1 5 x... 5 2 for each n. Now. note that l 1 1+1" _ 1+In_1 ﬂ lxu — xn—li " (1+x.)u+x.._o < ixn—xn—li 3% "' (1 + 1)(1 + 1) for each n = 2, 3, . . . . By Problem 4.15, the sequence {.15.} converges. Let held -314 = I ixn 4 xu—li x" w) x. Sineexn a 1 foreachn.weseeﬂ1atx 2 E. Then i )=1+1+I. x = linen“ =.1::e(1 + i..." That is, x is the positive solution of the equationx = 1+ i . or x2+x = 1+x+ 1. This implies J:2 = 2. and so I = 1/5. Problem 4.18. Deﬁne the sequence [xn] by I. = l and x,,+1=%(x,,+—2—-). r1:1.2,.... J:n Show that [1,.) converges and that lire 1,. = «5. Solution. Clearly, .15.. > 0 holds for each 1:. (Use induction to prove this!) Also, and so!) < x,,.H < x" holds for each n 2 2. By Theorem 4.3,x m limx" exism. Since 13' 2 2 holds for each n z 2, we see that x > 0. From the recursive formula, it follows that 21 = x + f, or x2 = 2. (Note also that the limit is independent of the iniu'al choice I] > 0.) Problem 4.19. Deﬁne the sequence x" = ELI % ﬁg:- n = l, 2, . , . . Show that {xn} does no! converge in IR. (See also Problem 5.10.) Solution. The inequality “"1... .J_ L Ina—1'" — n+1 +n+1+ +n+n 1 l _ lei 35+E+”'+ﬁ—"'h“2 shows that {In} is not a Cauchy sequence. and hence, is not convergent in R. ...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern