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Chapter 4: Problem 27

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

Short Answer

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

Step by step solution

01

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.
02

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 \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \).
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Suppose that \(\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=p .\) For which values of \(p\) must \(\sum_{n=1}^{\infty} 2^{n} a_{n}\) converge?

The following alternating series converge to given multiples of \(\pi .\) Find the value of \(N\) predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum \(N\) for which the error bound holds, and give the desired approximate value in each case. Up to 15 decimals places, \(\pi=3.141592653589793 .\)[T] \(\frac{\pi}{4}=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{2 n+1}\), error \(<0.0001\)

For the following exercises, indicate whether each of the following statements is true or false. If the statement is false, provide an example in which it is false.Suppose that \(a_{n}\) is a sequence of positive real numbers and that \(\sum_{n=1}^{\infty} a_{n}\) converges. Suppose that \(b_{n}\) is an arbitrary sequence of ones and minus ones. Does \(\sum_{n=1}^{\infty} a_{n} b_{n}\) necessarily converge?

Use the root test to determine whether \(\sum_{m=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is as follows. $$ a_{k}=\left(\frac{2 k^{2}-1}{k^{2}+3}\right)^{k} $$

Use the root test to determine whether \(\sum_{m=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is as follows. $$ a_{k}=\frac{1}{(1+\ln k)^{k}} $$
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