Question

Limits of sequences: Determine whether the sequences that follow are bounded, monotonic and/or eventually monotonic.

Determine whether each sequence converges or diverges. If the sequence converges, find its limit.

$\left\{\frac{e^k}{k}\right\}$

Short answer

The sequence is an increasing function with no upper bounds and the limit of the sequence does not exist.

Step-by-step solution

Step 1. Given Information

The given sequence is$\left\{\frac{e^k}{k}\right\}$.

Step 2. Use Derivative Test

• Consider the function $\frac{e^x}{x}$.
• Its derivative is $\frac{e^x(x-1)}{x^2}$.
• The derivative is positive when $x>1$.
• So, the function is an increasing function.

Step 3. Check Boundedness and Convergence

• The derivative test tells that the function has the minimum value at $x=1$.
• However, the function has no upper bounds.
• An unbounded function is non-convergent and hence, the limit of the sequence does not exist.