Question

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: If the tail of a sequence converges, the sequence converges.

(b) True or False: If $\left\{a_k\right\}$and $\left\{b_k\right\}$are two divergent sequences, then the sequence $a_k+b_k$diverges.

(c) True or False: If $\left\{a_k\right\}$is a convergent sequence of rational numbers, it must converge to a rational number.

(d) True or False: Every convergent sequence is bounded.

(e) True or False: Every bounded sequence is convergent.

(f) True or False: $\underset{k\to \infty }{lim}\frac{k^{1000000}}{{(1.000001)}^k}=0$

(g) True or False: Every increasing sequence of negative numbers converges.

(h) True or False: Every increasing sequence of positive numbers converges.

Short answer

(a) True.

(b) False.

(c) False.

(d) True.

(e) False.

(f) True.

(g) True.

(h) False.

Step-by-step solution

Part (a) Step 1. Given Information.

The tail of the sequence converges.

Part (a) Step 2. The definition.

For any $\in >0$, there exists some N > 0 such that if k > N, $\left|a_k-L\right|<\in fork\ge N$.

By definition, only the $\in$terms determines the convergence of the sequence. And this refers to the tail of the sequence.

So the statement is true.

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

Consider the divergent sequence,

$$\begin{gathered} \left\{a_k\right]=\{-k) \\ \left\{b_k\right\}=\left\{k\right\} \end{gathered}$$

The sum of two divergent sequence $\{a_k+b_k\}=\left\{0\right\}$which is a constant sequence, which is monotonic, bounded and convergent.

So the sum of two sequence may not be divergent.

So the given statement is false.

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

Consider the convergent sequence,

$\left\{a_k\right\}=\left\{{(1+\frac{1}{k})}^k\right\}$which is rational and convergent.

Now, the limit is,

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

So the limit is irrational.

So the limit of the rational sequence may be irrational.

So the given statement is false.

Part (d) Step 1. Determine true or false.

We know the theorem that,

If a sequence converges, then it is bounded.

So the given statement is true.

Part (e) Step 1. Consider the sequence.

Consider the sequence,

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

which is bounded and oscillatory but is not convergent.

So the given statement is false.

Part (f) Step 1. Use the result.

For any real numbers, $a>0,b>1,$the sequence $\left\{\frac{k^a}{b^k}\right\}$converges to zero.

So, $\underset{k\to \infty }{lim}\frac{k^{1000000}}{{(1.000001)}^k}=0$is true statement.

Part (g) Step 1. Use the result.

The monotonic increasing sequence is convergent if it is bounded above.

From the statement, the increasing sequence of the negative numbers is always bounded by the first term of the sequence.

So every increasing sequence of negative numbers converges.

So the given statement is true.

Part (h) Step 1. Use the result.

The monotonic increasing sequence is convergent if it is bounded above.

From the statement, the increasing sequence of the positive numbers is not always bounded by the first term of the sequence. so it is not convergent.

So every increasing sequence of positive numbers not converges.

So the given statement is false.