Q. 22 - Calculus

Question

Discuss the boundedness and monotonicity of the geometric sequence ${\left\{c{r}^k\right\}}_{k=0}^{\infty }$ with $c>0$ and $r<0$. In addition, determine whether the sequence converges or diverges. If it converges, find the limit of the sequence.

Step-by-Step Solution

Answer

The sequence is not a monotonic sequence, unbounded and divergent.

Step 1. Given information

We have to tell about the boundedness and monotonicity of the sequence ${\left\{c{r}^k\right\}}_{k=0}^{\infty }$ with $c>0$ and $r<0$.

Step 2. Determine the monotonicity and boundedness also determine whether the sequence converges or diverges.

$a_k=c{r}^k$

The ratio $r$ is negative because it is given $r<0$.

As the index $k$ increases, the sign of the sequence varies.

Thus, the sequence is not a monotonic sequence.

The given sequence has neither upper bound nor lower bound.

Therefore, the sequence is unbounded.

The sequence is neither monotonic nor bounded and hence cannot be convergent.

Therefore, the sequence is divergent.