Sheet 4 Moodle version

This is the same content as the pdf in the Sheet 4 assignment, but visible as a Moodle page.

Q1

(a) Let the sequence \( (a_n) \) be defined by

\[ a_n = \left(\frac{n^2-1}{n^2+1}\right)\cos\left(\frac{2n\pi}{3}\right). \]

By considering suitable subsequences, prove that \( (a_n) \) diverges.

(b) Consider the sequence \( (\cos n) \).  Show that, for a suitable positive constant \( K \), there exist subsequences \( (b_r) \) and \( c_s) \) of \( (\cos n) \) with \( b_r > K \) for all \( r \) and \( c_s < -K \) for all \( s \).  Deduce that \( (\cos n) \) diverges.

Q2

(a) Let \( (a_n) \) be a sequence such that the subsequences \( (a_{2n}) \) and \( (a_{2n+1}) \) both converge to a real number \( L \).  Show that \( (a_n) \) also converges to \( L \).

(b) Let \( (b_n) \) be a sequence such that each of the subsequences \( (b_{2n}) \), \( (b_{2n+1}) \), \( (b_{3n}) \) converges.  Need \( (b_n) \) converge? Give a proof or a counterexample.

(c) Let \( (c_n) \) be a sequence such that the subsequence \( (c_{kn}) \) converges for each \( k = 2, 3, 4, \dotsc \).  Need \( (c_n) \) converge?  Give a proof or a counterexample.

Q3

For each of the following choices of \( a_n \), determine whether the sequence \( (a_n) \) converges.  Justify your answers, and find the value of the limit when it exists.

1. \( \displaystyle \frac{n^2}{n!}; \)
2. \( \displaystyle \frac{2^n n^2 + 3^n}{3^n(n+1)+n^7}; \)
3. \( \displaystyle \frac{(n!)^2}{(2n)!}; \)
4. \( \displaystyle \frac{n^4 + n^3 \sin n + 1}{5n^4 - n \log n}.\)

[You may freely make use of standard limits and inequalities, Sandwiching and the Algebra of Limits, as appropriate.]

Q4

(a) Let \( (a_n) \) be a real sequence, and assume that \( a_n \geq 0 \) and \( a_n \to L \).  Prove from the limit definition that \( L\geq 0 \).  Prove further that \( \sqrt{a_n} \to \sqrt{L} \).

(b) Let \( (a_n) \), \( (b_n) \) and \( (c_n) \) be sequences of real numbers converging to \( L_1 \), \( L_2 \), \( L_3 \) respectively.  Let \( d_n = \max\{a_n,b_n,c_n\} \).  Assuming any standard AOL results that you require, prove that \( d_n \to \max\{L_1,L_2,L_3\} \).

Q5

Let \( r > 0 \).  Let \( a_n = \displaystyle \frac{r^n}{n!} \).

(a) By considering \( \displaystyle \frac{a_{n+1}}{a_n} \) show that the tail \( (a_n)_{n \geq N} \) is monotonic decreasing if \( N \) is sufficiently large.  [You should specify a suitable value of \( N \).]

(b) Show that \( (a_n) \) converges to a limit \( L \) and find the value of \( L \).

Q6

The real sequence \( (a_n) \) is defined by

\[ a_1 = c, \quad (\alpha + \beta)a_{n+1} = a_n^2 + \alpha\beta, \]

where \( 0 < \alpha < \beta \) and \( c > \alpha \).

(a) Prove that if \( (a_n) \) converges to a limit \( L \) then necessarily \( L = \alpha \) or \( L = \beta \).

(b) Prove that \( a_{n+1} - \gamma \) and \( a_n - \gamma \) have the same sign, where \( \gamma \) denotes either \( \alpha \) or \( \beta \).

(c) Prove that if \( c < \beta \), then \( (a_n) \) converges monotonically to \( \alpha \).  Discuss the limiting behaviour of \( (a_n) \) when \( c \geq \beta \).

(d) Prove that if \( \alpha < c < \beta \), then

\[ |a_n - \alpha| \leq \left(\frac{\alpha+c}{\alpha+\beta}\right)^{(n-1)}(c-\alpha). \]

Q7

(Optional, to provide additional practice with sequences defined by recurrence relations.)  Let \( (a_n) \) be the sequence of real numbers given by

\[ a_1 = a, \quad a_{n+1} = \frac{2}{a_n+1} \quad (n \geq 1). \]

(a) Assume that \( 0 < a < 1 \).  Prove that the subsequences \( (a_{2n}) \) and \( (a_{2n+1}) \) are monotonic, one increasing and the other decreasing.  Prove that each of these subsequences converges, and find their limits.  Deduce that \( (a_n) \) converges.

(b) What happens if \( a > 1 \)?