1549901369-Elements_of_Real_Analysis__Denlinger_

2.5 Monotone Sequences 105

2. Which of the following sequences are eventually monotone (or strictly) in-
   creasing (or decreasing)? Justify your answers, assuming the usual prop-
   erties of trigonometric functions where necessary.

(a) {(-~)n}

(c) {n+ (-~)n}

(e) {2 + (-l)n}

(g) {sin n

2

7r }

(i) { n }

2n

(k) {cos 2:}

(m) { n2

3

:: ~ 2}

3. Prove Theorem 2.5.3 (b).


4. Prove Corollary 2.5.4.

(d) {n^2 - lOn + 100}

(f) {3n

2

+n(-1r}

(h) {sin n?r}

(j) { ~~}

(1) {sin

3

:}

(n) {Jn- k}

5. The proof of Theorem 2.5.3 (monotone convergence theorem) depends
   heavily on the completeness property of R Find a bounded, monotone
   increasing sequence in the (incomplete) field Q that fails to converge in
   Q.


6. Consider the sequence {xn} defined inductively by x 1 = 1, and Vn EN,
   Xn+l = y'6 + Xn· Prove that {xn} converges, and find its limit. [Hint: See
   Example 2.5.8.J


7. Consider the sequence {xn} defined inductively by x 1 = 1, and Vn EN,
   Xn+l = y'4xn + 5. Prove that {xn} converges, and find its limit. [Hint:
   See Example 2.5.8.J


8. Consider the sequence {xn} defined inductively by x 1 = 1, and Vn EN,
   xn+l =

2

xn

7

+

3

. Prove that { xn} converges, and find its limit.
9. Consider the sequence {xn} defined inductively by x 1 = 1, and Vn E N,
2.
Xn+i = nxn. Prove that { Xn} converges, and find its limit.
n+l
10. Consider the sequence {xn} defined inductively by x 1 = c E JR, and
Vn E N, Xn+i = x;;,. For what values of c does {xn} converge, and to
what?