Question

Some Convergent Sequences Involving Exponents: For any real number p > 0, the following sequences converge. Fill in each blank with the appropriate value.

${\left(\frac{1}{k}\right)}^p\to$

Short answer

The required answer is${\left(\frac{1}{k}\right)}^p\to 0$

Step-by-step solution

Step 1. Given information   

The given data is for any real numberp > 0, the sequences converge.

Step 2. Explanation     

The sequence $\frac{1}{k}$is a decreasing sequence as k increases $\frac{1}{k}$decreases.

For p>0, for increasing k, the terms of the given sequence decreases.

Thus, the given sequence ${\left(\frac{1}{k}\right)}^p$decreases.

Hence, ${\left(\frac{1}{k}\right)}^p\to 0$as it is a null sequence.