116 Chapter 2 • Sequences- Suppose {xn} is a bounded sequence. Prove that if all its convergent
subsequences have the same limit, L, then { Xn} also converges, and has
limit L. - Use Exercise 6 to prove that lim \inf = +oo. Hint: first show that
n->oo
nn < (2n)!, from which you can get^2 .y!(2"n)T > fo. Then show that
(2n + 1 )! > nn+i, from which you can get^12 2 n+y'(2n + 1)! > fo. - (Project) Recursive Arithmetic Means:^13 Let a,b > 0. Define {xn}
Xn+ l + Xn.
by x 1 = a, x 2 = b, and Vn E N, Xn+2 =
2
. That is , every term·
beginning with the third is t he arithmetic mean of the preceding two
terms.
(a) By writing out eight or ten terms, conj ecture a formula for the odd-
numbered terms, and a formula for the even-numbered terms. Specif-
ically, find formulas for r n and Sn such that Vn E N,
X2n-l = b + r2n-l (a - b) and X2n = b + S2n(a - b).
(b) Use mathematical induction to prove the formulas for rn and Sn
conjectured in (a). [Hint: prove both at the same time.]
(c) Show that one of the sequences {x2n-d and {x2n} is monotone
increasing and the other is monotone decreasing.
(d) P rove that both {x2n+d and {x2n} converge, and to the same limit.
(e) Use Exercise 6 to determine lim Xn·
n->oo24. (Project) Recursive Geometric Means: Let a, b > 0. Define { Xn}
by X1 = a, X2 = b, and Vn E N, Xn+2 = Jxn+1Xn· That is, every term
beginning with the third is the geometric mean of the preceding two
terms. Repeat the instructions (a)-(e) of Exercise 23, but this time show
that Vn EN,(
a) r2n-1 (a) s2n
X2n+l = b b and X2n = b bwhere rn and Sn are the same as used in Exercise 23.25. (Project) The Arithmetic-Geometric Mean of Two Positive N um-
bers: Let 0 <a< b. Define two sequences {xn} and {yn} inductively by
a+ b CL Xx+ Yn
X1 = -
2- , Y1 = vab, and Vn EN, Xn+i =
2
, and Yn+l = vfXnYn·
You may assume that Vm < n in N, and x > 1, 'i;:!X > iy'X.
For an easier derivation of the limit of t his sequence, see Exercise 2.7.9.