Chapter 1: Problem 18

Use the ratio test to find whether the following series converge or diverge: 18\. \(\sum_{n=1}^{\infty} \frac{2^{n}}{n^{2}}\)

Short Answer

Expert verified

The series diverges.

Step by step solution

Define the given series

The series in question is \(\sum_{n=1}^{\infty} \frac{2^{n}}{n^{2}}\)

Apply the ratio test

The ratio test involves finding the limit \(L = \lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| \), where \( a_n = \frac{2^{n}}{n^{2}} \).

Compute \(\frac{a_{n+1}}{a_n}\)

Let's compute \(\frac{a_{n+1}}{a_n} \):\( a_{n+1} = \frac{2^{n+1}}{(n+1)^{2}} \), so \( \frac{a_{n+1}}{a_n} = \left( \frac{2^{n+1}}{(n+1)^{2}} \right) \div \left( \frac{2^{n}}{n^{2}} \right) = \frac{2 \cdot 2^n}{(n+1)^2} \cdot \frac{n^2}{2^n} = \frac{2n^2}{(n+1)^2}\)

Simplify the expression

Simplify \(\frac{2n^2}{(n+1)^2}\):\( \frac{2n^2}{(n+1)^2} = 2 \cdot \frac{n^2}{n^2 \left( 1 + \frac{1}{n} \right)^2} = 2 \cdot \frac{1}{\left( 1 + \frac{1}{n} \right)^2} \). As \( n \rightarrow \infty\), \(\frac{1}{n} \rightarrow 0\), so the limit becomes \(2 \cdot \frac{1}{1^2} = 2\).

Determine the behavior of the limit

Compare limit \(L = 2\) with 1. Since \(L = 2 > 1\), according to the ratio test, the series \(\sum_{n=1}^{\infty} \frac{2^{n}}{n^{2}}\) diverges.

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Key Concepts

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

series convergence

Series convergence is a fundamental concept in calculus and mathematical analysis. When we talk about a series, we refer to the sum of the terms of a sequence. The series can either converge or diverge. A series converges if the sum of its terms approaches a specific value as the number of terms grows indefinitely. Conversely, if the series does not settle to a fixed value, it diverges.

There are several methods to test for convergence, including the Ratio Test, which we used in the provided exercise. Knowing whether a series converges is crucial because it tells us if the sum has a finite value. This is particularly important for applications in physics, engineering, economics, and many other fields.

limit comparison test

The Limit Comparison Test is another useful tool to determine the convergence or divergence of a series. This test compares the terms of our series with the terms of a known benchmark series. Here's how it works:

Given two series \(\boldsymbol{\sum a_n}\) and \(\boldsymbol{\sum b_n}\):

Find \(\boldsymbol{\lim_{n \to \infty} \frac{a_n}{b_n}}\).
If this limit is a finite positive number, both series will either converge or diverge together.

This test is especially helpful when the terms of one series are complex, but closely resemble the terms of another series of known behavior. It simplifies the evaluation process by leveraging our understanding of simpler benchmark series.

infinite series

An infinite series sums an endless sequence of terms. Not all infinite series produce a finite sum. Understanding the behavior of infinite series is critical in various branches of mathematics and applied sciences.

Important concepts to know about infinite series include:

Convergence: An infinite series converges if the sum of its infinite terms approaches a specific, finite number.
Divergence: If the series does not approach a specific value, it is divergent.
Partial sums: These are the sums of the first \(n\) terms of the series, which help evaluate the series' overall behavior.
Tests for convergence: Various tests, like the Ratio Test and Limit Comparison Test, help determine if an infinite series converges or diverges.

In our exercise, we used the Ratio Test to determine that the given series \(\sum_{n=1}^{\infty} \frac{2^n}{n^2}\) diverges, highlighting how these concepts play out in practical scenarios.