Question

Verify that the Ratio Test is inconclusive for the \(p\) -series. $$ \sum_{n=1}^{\infty} \frac{1}{n^{p}} $$

Short answer

The Ratio Test is inconclusive for the \( p \) -series.

Step-by-step solution

Define the series

The \(p\)-series is given by \( \sum_{n=1}^{\infty} \frac{1}{n^{p}} \). We need to apply the Ratio Test to this series to check its convergence.

Apply the Ratio Test

The Ratio Test involves finding the limit of \( \frac{a_{n+1}}{a_{n}} \), where \( a_{n} \) are the terms of the series. In this case, the series is \( \frac{1}{n^{p}} \), hence, \( a_{n} \) is \( \frac{1}{n^{p}} \) and \( a_{n+1} \) is \( \frac{1}{(n+1)^{p}} \). We then compute the limit as \( n \) approaches infinity.

Compute the limit

Computing the limit, we get \( \lim_{n\to\infty} \frac{(n^{p})}{(n+1)^{p}} = \lim_{n\to\infty} \frac{1}{(1+\frac{1}{n})^{p}} \). After simplifying, the limit equals to 1.

Interpret the result

Since the limit obtained from the Ratio Test equals to 1, the Test is inconclusive, i.e., it cannot determine whether the \( p \) -series converges or diverges.