Question

Determine the convergence or divergence of the sequence with the given \(n\) th term. If the sequence converges, find its limit. \(a_{n}=\frac{n^{p}}{e^{n}}, p>0\)

Short answer

The sequence converges and its limit is 0.

Step-by-step solution

Apply the Limit Test

To determine whether a sequence converges, check if the limit of the sequence as \(n\) approaches infinity exists. In other words, calculate \(\lim_{n\to\infty}\frac{n^{p}}{e^{n}}\).

Application of L'Hopital's rule

Given that it's the limit of type \(0/0\) or \(\infty/\infty\), apply L'Hopital's rule. The rule states that \(\lim_{x\to c}\frac{f(x)}{g(x)}\) is equivalent to \(\lim_{x\to c}\frac{f'(x)}{g'(x)}\) when either limit involves an indeterminate form.

Finding the derivative

The derivative of \(n^{p}\) is \(p \cdot n^{p-1}\) and the derivative of \(e^{n}\) is \(e^{n}\). Therefore, the new limit becomes \(\lim_{n\to\infty}\frac{p \cdot n^{p-1}}{e^{n}}\). Since the denominator increases exponentially faster than the numerator, this limit equals zero.

Determine convergence or divergence

A sequence converges if the limit exists and is finite. In this case, the limit is 0, hence the sequence converges.