Q. 41 - Calculus

Question

Determine whether the sequence is monotonic or eventually monotonic and whether the sequence is bounded above and/or below. If the sequence converges, give the limit. 

    

   $\left\{\left(42+(-1{)}^k\right)\right\}$

Step-by-Step Solution

Answer

Ans:  The nonmonotonic decreasing sequence $\left\{a_k\right\}=\left\{\left(42+(-1{)}^k\right)\right\}$ is bounded and hence is not convergent. Therefore, the sequence is not convergent.

Step 1. Given information.

given,

     $\left\{\left(42+(-1{)}^k\right)\right\}$

Step 2. The objective is to determine whether the sequence is monotonic, bounded above, or bounded below, and to find the limit of the sequence if the sequence is convergent.

  In the sequence $\left\{a_k\right\}=\left\{\left(42+(-1{)}^k\right)\right\}$ the general term is $a_k=42+{(-1)}^k$.

Step 3. The general term of the sequence is a k = 42 + ( - 1 ) k .

The term $a_{k+1}-a_k$ gives,

  $\begin{array}{r}{a}_{k+1}-a_k=\left(42+(-1{)}^{k+1}\right)-\left(42+(-1{)}^k\right) \text{ (Substitution) }\\ =42+(-1{)}^{k+1}-42-(-1{)}^k \\ =(-1{)}^k(-1)-(-1{)}^k \\ \left.=-(-1{)}^k-(-1{)}^{k }\text{ (Factorize }\right)\\ =-2(-1{)}^k \end{array}$

The term on the right-hand side is positive for the odd value of k and is negative for the even value of k. Thus, the sequence is neither increasing nor decreasing.

The sequence $\left\{a_k\right\}=\left\{\left(42+(-1{)}^k\right)\right\}$ is not monotonic. The given sequence is not monotonic.

Step 4. Then,

The sequence $\left\{a_k\right\}=\left\{\left(42+(-1{)}^k\right)\right\}$is bounded because terms of the given sequence are alternatively $41$ and $43$ .

$41\le a_1\le 43\text{ for }k>0$

The given sequence has lower and upper bounds, therefore, the sequence is bounded.

Step 5. Thus,

The nonmonotonic decreasing sequence $\left\{a_k\right\}=\left\{\left(42+(-1{)}^k\right)\right\}$ is bounded and hence is not convergent. Therefore, the sequence is not convergent.