Question

State whether the given \(p\) -series converges. \(\sum_{n=1}^{\infty} \frac{1}{n \sqrt{n}}\)

Short answer

The series converges because it is a p-series with \( p = 3/2 \), which is greater than 1.

Step-by-step solution

Identify the series as a p-series

The given series is \( \sum_{n=1}^{\infty} \frac{1}{n \sqrt{n}} \). A p-series generally has the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). We need to express the given series in the form of a p-series.

Rewrite the series in terms of p

Rewrite the term \( \frac{1}{n \sqrt{n}} \) by recognizing that \( \sqrt{n} = n^{1/2} \). Therefore, \( \frac{1}{n \sqrt{n}} \) can be rewritten as \( \frac{1}{n^{3/2}} \). This means that the series is \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \).

Apply the p-series test

The p-series \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) converges if \( p > 1 \) and diverges if \( p \leq 1 \). For our series, \( p = 3/2 \). Since \( 3/2 > 1 \), the series converges.

Key concepts

p-series test

One of the most reliable tests in determining the convergence of series is the p-series test. It is specifically used for series that can be written in the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), where \( n \) is a positive integer and \( p \) is a real number.

The key characteristic of the p-series test is centered around the value of \( p \):

• If \( p > 1 \), the series converges, meaning it adds up to a finite number.
• If \( p \leq 1 \), the series diverges, implying that the sum of its terms tends to infinity or does not settle at a particular value.

This test is extremely useful as it provides a straightforward method to quickly assess whether a series will converge or not. In the analyzed exercise, the series \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \) has \( p = \frac{3}{2} \), which is greater than 1. Therefore, we conclude the series converges.

series convergence

Series convergence is a central concept in calculus, where we determine whether the sum of infinitely many terms adds up to a finite number. In more practical terms, convergence implies that as you keep adding more terms in the series, they approach a specific limit.

Numerous tests identify convergence, but they all boil down to analyzing how terms behave as they progress indefinitely:

• The terms should get smaller as the series advances.
• There should be a rule or pattern to these terms that dictates their inclusion in the series.

In the case of a p-series, convergence largely depends on the value of \( p \). Understanding series convergence allows mathematicians and students alike to predict outcomes and behaviors of functions represented as sums, a powerful tool in both theoretical and applied mathematics.

mathematical series

A mathematical series is essentially the sum of the terms of a sequence. It is a way to express complex summations in a compact form. Series are often used to represent functions or solve complex equations involving infinite sums.

To further understand:

• A series can be finite or infinite. A finite series has a definite, countable number of terms.
• An infinite series continues indefinitely, potentially with terms that follow a rule or pattern.
• Different types of series include arithmetic series, geometric series, and the p-series among others.

The concept of a series is fundamental in exploring topics like calculus and advanced mathematics. Recognizing a series form, such as the p-series, can dramatically simplify the process of finding whether a sum converges and what methods to apply for easy computation.