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
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![### Understanding the Ratio Test for Series Convergence
The 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
\[ \sum_{k=1}^{\infty} \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87a5eb21-df18-43d0-b53f-a372e6adca02%2Fdf8e4b8c-4498-47ff-a53f-0e2e11697ac3%2Fc6hm1rn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding the Ratio Test for Series Convergence
The 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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