106 Chapter 2 • Sequences
11. Consider the sequence {xn} defined inductively by x 1 = 1, and \:Jn E N,
Xn+i = ~Xn + 6. Prove that {xn} converges, and find its limit.
1
12. Define {xn} inductively by X1 = 1 and Xn+l = Xn +-.Prove that {xn}
Xn
is monotone increasing, but diverges.
1
13. Define {xn} inductively by x 1 = 2 and Xn+l = 2--. Write out the first
Xn
few terms of this sequence and conjecture a formula for Xn. Prove this
formula by mathematical induction, and use it to find lim Xn·
n->oo{1 · 3 · 5 · 7 · · · · · (2n - 1) }
1 4. Prove that ( ) converges.^10
2 · 4 · 6 · 8 · · · · · 2n
Use the method of Example 2.5. 12 to calculate .JIO to four decimal places.
Use the method of Example 2 .5.12 to calculate J45 to five decimal places.
Use the methods of this section to give an easy proof of Theorem 2.3.7.
Prove Theorem 2.5.13 (b).
19. Prove Theorem 2.5.14 (b).. n 1.
20. About how large 1s Sn = L k when n = 500, 000?
21.22.k=l
n 1
\:Jn EN, let Sn = ~ k(k + l). Prove that {Sn} converges, and find itslimit. Hint: Find an easy formula for Sn by writing it out as a sum and
using the "partial fraction decomposition" of k(k ~ l)n 1
\:Jn E N, let Tn = L k 2. Prove that {Tn} converges. [Hint: use the
k=l
monotone convergence theorem combined with Exercise 21.]- Prove or disprove (if false, give a counterexample.):
(a) If {xn} and {yn} are both monotone increasing, then so is their sum
{xn + Yn}·
(b) If {xn} and {Yn } are both monotone increasing, then so is their
difference { Xn -Yn}. - In fact, the limit is 0 , but the proof is postponed until Exercise 8.2.44.