### Understanding the Ratio Test for Series ConvergenceThe Ratio Test is an essential tool in determining the convergence or divergence of an infinite series. Let's consider a sequence \(\{a_k\}\) where \(a_k \neq 0\) for all \(k\). The test involves the limit\[ \ell = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|. \]The conclusions drawn from this limit \(\ell\) are as follows:1. If \(\ell < 1\), the series \(\sum a_k\) converges absolutely.2. If \(\ell > 1\), the series \(\sum a_k\) diverges.3. If \(\ell = 1\), the test is inconclusive.To explore the inconclusiveness when \(\ell = 1\), we need to demonstrate examples of both a convergent series and a divergent series where \(\ell = 1\).#### Example of a Convergent Series with \(\ell = 1\)Consider the series\[ \sum_{k=1}^{\infty} \frac{1}{k^2}. \]For this series,\[ a_k = \frac{1}{k^2}. \]Calculating the limit,\[ \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = \lim_{k \to \infty} \left| \frac{\frac{1}{(k+1)^2}}{\frac{1}{k^2}} \right| = \lim_{k \to \infty} \left| \frac{k^2}{(k+1)^2} \right| = \lim_{k \to \infty} \left( \frac{k}{k+1} \right)^2 = 1. \]Thus, \(\ell = 1\). However, we know that the series \(\sum \frac{1}{k^2}\) is convergent by the p-test (p-series with \(p > 1\)).#### Example of a Divergent Series with \(\ell = 1\)Now consider the harmonic series\[ \sum_{k=1}^{\infty} \

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The Ratio Test involves a sequence {ar} such that ar +0 for all k and l = lim ak+1| Show that The Ratio Test fails when e= 1. That is, give examples of: |3D a convergent series ar such that e= 1; and • a divergent series ar such that = 1. |3D

The Ratio Test involves a sequence {ar} such that ar +0 for all k and l = lim ak+1| Show that The Ratio Test fails when e= 1. That is, give examples of: |3D a convergent series ar such that e= 1; and • a divergent series ar such that = 1. |3D

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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