2) of the Ratio test Prove Case If lime | an+ 1 |>1 an 1 1 - Proof: Assume lim 7380 *Sentence based Then an Diverges E antal leng an = Limit >1. or =100
2) of the Ratio test Prove Case If lime | an+ 1 |>1 an 1 1 - Proof: Assume lim 7380 *Sentence based Then an Diverges E antal leng an = Limit >1. or =100
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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I need help with providing a Proof of Divergence in the Ratio Test with an explanation of how it is proved. I need it explained both verbally and mathematically. It's described below:
![**Prove Case 2 of the Ratio Test**
*Sentence Based*
If
\[
\lim_{{n \to \infty}} \left| \frac{{a_{n+1}}}{a_n} \right| > 1
\]
then
\[
\sum a_n
\]
diverges.
**Proof:**
Assume
\[
\lim_{{n \to \infty}} \left| \frac{{a_{n+1}}}{a_n} \right| = \text{Limit} > 1
\]
or \(\infty\).
**Explanation:** The Ratio Test involves evaluating the limit of the absolute value of the ratio of consecutive terms in a series. If this limit is greater than 1, the series diverges. In the proof, we start by assuming the condition given in the test and proceed logically from there.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e135d5c-f867-4afd-9f0d-b505f4f19664%2F96404a6d-acd2-4b94-92d8-ef41ad95f266%2Flgecyh6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Prove Case 2 of the Ratio Test**
*Sentence Based*
If
\[
\lim_{{n \to \infty}} \left| \frac{{a_{n+1}}}{a_n} \right| > 1
\]
then
\[
\sum a_n
\]
diverges.
**Proof:**
Assume
\[
\lim_{{n \to \infty}} \left| \frac{{a_{n+1}}}{a_n} \right| = \text{Limit} > 1
\]
or \(\infty\).
**Explanation:** The Ratio Test involves evaluating the limit of the absolute value of the ratio of consecutive terms in a series. If this limit is greater than 1, the series diverges. In the proof, we start by assuming the condition given in the test and proceed logically from there.
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