Question

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

$p^{\frac{1}{k}}\to$

Short answer

The required answer is$p^{\frac{1}{k}}\to 1$

Step-by-step solution

Step 1. Given information    

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

Step 2. Explanation    

As we know that the sequence $\frac{1}{k}$converges to zero. Since the exponential function $p^x$is continuous for p>0.

Thus, by using limits of function of sequence, which states that if $a_k\to L$ and f is a function that is continuous at L then $f\left(a_k\right)\to f\left(L\right)$

Thus, the sequence $p^{\frac{1}{k}}$converges to $p^0=1$

Hence,$p^{\frac{1}{k}}\to 1$