2. Prove the limit ratio test (Corollary 3.8 in the notes): Suppose {an} is a sequence of nonzero real numbers and r(n) = |"n+1|. Suppose further that lim,+ r(n) = L. Show that (i) If L < 1, the series an converges absolutely. (ii) If L > 1, the series an diverges. Hint. This is a Corollary to the ratio test (Theorem 3.6 in the notes), so you should be using the ratio test in your proof. n→∞
2. Prove the limit ratio test (Corollary 3.8 in the notes): Suppose {an} is a sequence of nonzero real numbers and r(n) = |"n+1|. Suppose further that lim,+ r(n) = L. Show that (i) If L < 1, the series an converges absolutely. (ii) If L > 1, the series an diverges. Hint. This is a Corollary to the ratio test (Theorem 3.6 in the notes), so you should be using the ratio test in your proof. n→∞
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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![2. Prove the limit ratio test (Corollary 3.8 in the notes): Suppose \(\{a_n\}\) is a sequence of nonzero real numbers and
\[
r(n) = \left| \frac{a_{n+1}}{a_n} \right|.
\]
Suppose further that \(\lim_{n \to \infty} r(n) = L\). Show that
(i) If \(L < 1\), the series \(\sum a_n\) converges absolutely.
(ii) If \(L > 1\), the series \(\sum a_n\) diverges.
*Hint*: This is a Corollary to the ratio test (Theorem 3.6 in the notes), so you should be using the ratio test in your proof.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32b90586-0d73-43dc-903b-9a80d199c8ec%2Fd6ccee62-d522-492f-a6ea-5714730103a3%2Fkqa537a_processed.png&w=3840&q=75)
Transcribed Image Text:2. Prove the limit ratio test (Corollary 3.8 in the notes): Suppose \(\{a_n\}\) is a sequence of nonzero real numbers and
\[
r(n) = \left| \frac{a_{n+1}}{a_n} \right|.
\]
Suppose further that \(\lim_{n \to \infty} r(n) = L\). Show that
(i) If \(L < 1\), the series \(\sum a_n\) converges absolutely.
(ii) If \(L > 1\), the series \(\sum a_n\) diverges.
*Hint*: This is a Corollary to the ratio test (Theorem 3.6 in the notes), so you should be using the ratio test in your proof.
![**Theorem 3.6 (Ratio Test).** Let \( a_n \) be a sequence of nonzero real numbers and let
\[
r(n) = \left| \frac{a_{n+1}}{a_n} \right|.
\]
(i) If there exists \( c < 1 \) so that \( r(n) \leq c \) for all \( n \), then the series \( \sum a_n \) converges absolutely.
(ii) If \( r(n) \geq 1 \) for all \( n \), then the series \( \sum a_n \) diverges.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32b90586-0d73-43dc-903b-9a80d199c8ec%2Fd6ccee62-d522-492f-a6ea-5714730103a3%2F0tnzv18_processed.png&w=3840&q=75)
Transcribed Image Text:**Theorem 3.6 (Ratio Test).** Let \( a_n \) be a sequence of nonzero real numbers and let
\[
r(n) = \left| \frac{a_{n+1}}{a_n} \right|.
\]
(i) If there exists \( c < 1 \) so that \( r(n) \leq c \) for all \( n \), then the series \( \sum a_n \) converges absolutely.
(ii) If \( r(n) \geq 1 \) for all \( n \), then the series \( \sum a_n \) diverges.
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