Q. 02 - Calculus

Question

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A sequence $\left\{a_k\right\}$ of irrational numbers that converges to a rational number.

(b) A bounded sequence without a limit.

(c) A convergent sequence that is not eventually monotonic.

Step-by-Step Solution

Answer

(a) $\left\{a_k\right\}=\left\{\frac{\sqrt{2}}{k}\right\}$

(b) A bounded sequence without a limit is $\left\{a_k\right\}=\{-1,1\}$.

(c) A convergent sequence that is not eventually monotonic is $\left\{a_k\right\}=\left\{\frac{{(-1)}^k}{k+6}\right\}$.

Part (a) Step 1. Given Information.

The sequence of irrational numbers.

Part (a) Step 2. Consider the convergent sequence.

Consider the convergent sequence of irrational numbers,

$$\begin{gathered} \left\{a_k\right\}=\left\{\frac{\sqrt{2}}{k}\right\} \\ \underset{k\to \infty }{lim}\left\{a_k\right\}=\underset{k\to \infty }{lim}\left\{\frac{\sqrt{2}}{k}\right\} \\ =0 \end{gathered}$$

which is a rational number.

Part (b) Step 1. Consider the convergent sequence.

Consider the convergent sequence,

$\left\{a_k\right\}=\{-1,1\}$

The sequence i bounded as $-1\le a_k\le 1$.

And it is oscillatory and it does not have any limit.

Part (c) Step 1. Consider the sequence.

Consider the sequence,

$\left\{a_k\right\}=\left\{\frac{{(-1)}^k}{k+6}\right\}$.

It is not a monotonic sequence since the sign of the sequence varies alternatively with k.

Part (c) Step 2. Check for bounded and convergent.

The sequence is a bounded sequence because it has the upper and lower bounds as $0\le a_k\le \frac{1}{7} for k>0$.

So the sequence is bounded.

$$\begin{gathered} \underset{k\to \infty }{lim}\left\{a_k\right\}=\underset{k\to \infty }{lim}\left\{\frac{{(-1)}^k}{k+6}\right\} \\ =0 \end{gathered}$$

So the sequence is bounded and convergent but not monotonic.