644 Appendix C • Answers & Hints for Selected Exercises- Modify the proof of (a), replacing "increasing" by "decreasing," "sup" by
"inf," etc. - {dn}, where dn is the decimal expansion of v'2 ton decimal places.
- First show {xn} monotone increasing, by math induction. Then, show {xn}
bounded above (by 5), also by induction. By Thm. 2.5.3, {xn} converges, say
to L. Then n--) lim ex> Xn+l = n-+ lim ex:> J4xn+5 L = J4L+5 L^2 = 4L+5
L^2 - 4L - 5 = 0 L = 5 or L = -1. By Thm. 2.3.12, L ~ 0 .. ·. L = 5. - First show by induction that Vn E N , 0 < Xn S 1. For the general step,
0 < Xk S 1 * 0 < x~ S Xk S 1, and since 0 < k!i < 1, it follows that
0 < k!i x~ s 1; i.e., 0 < Xk+l s 1. To show that {xn} is monotone decreasing,
... (k + l)x~+ 1 k + 1
the general mduct1on step is Xk+2 = k = -k--· Xk+l · Xk+1 <
+2 +2
Xk+l· By the monotone convergence thm., 3L = lim Xn. Then lim Xn+l
n-+CXJ n-+oo
n~~ r n+l n Xn^2 - - n~~ r n+l n. n~~ r Xn^2 - - L^2 , SO L^2 - L -- O· ,. Le., L --^0 or^1. s· mce
{xn} is decreasing, L =f. l, so L = 0. - Show by induction that {xn} is monotone increasing and bounded above,
say by 2, so by the monotone convergence thm., it converges. Then L =
n->oo lim Xn L = ijL + 6 L^3 -L-6 =^0 (L-2)(L^2 +2L+3) =^0 L = 2.
- Note that 0 < Xn+i = Xn · ~~$~ < Xn· Apply the monotone convergence
theorem. - Take X1 = 3 and calculate x 1 ,x 2 ,x 3 ,··· until x;1, < 10+ 1.5 x 10-^6. Now
X4 = 3.1622776 · · · and x~ = 10.0000000 · · ·, so to 4 decimal places, v'10 =
3.1623. - lal < 1 * {lanl} is strictly decreasing. By the monotone convergence thm.,
{Ian I}, L ~ 0. Taking limit of both sides of lan+ll = lal ·lain yields L = Llal,
so L(l - lal) = 0. Since lal =/. l, L = 0. :. lanJ , 0, so an__, 0. - Between 13. 12 and 14.12.
- Since k(k~l) = i-k!1' £:: k(k~l) = (1 - ~) + (~ - ~) + · · · (~ - n!1) =
k=l
l - n!l -7 1. - (a) True. Xn S Xn+i, Yn S Yn+l Xn + Yn S Xn+l + Yn+l·
(b) False. Take Xn = n + ~ and Yn = n.
(c) True. 0 S Xn S Xn+l, 0 S Yn S Yn+l 0 S XnYn S Xn+lYn+l·
(d) False. Take Xn = Yn = -~.
(e) False. Take Xn = n and Yn = n^2.
)