Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Instructions:**
Determine whether the series converges or diverges. If it converges, find the sum.
**Mathematical Expression:**
\[
\sum_{n=1}^{\infty} \left( \frac{6}{n^2 + 7n + 12} + 4^{-n-2} \cdot 2^n \right)
\]
**Explanation:**
This series contains two components:
1. **Fractional Term:** \(\frac{6}{n^2 + 7n + 12}\)
- This part represents a rational function where the numerator is constant and the denominator is a quadratic expression in \(n\).
2. **Exponential Term:** \(4^{-n-2} \cdot 2^n\)
- This part includes an exponential expression with a base of 4 raised to the power of \(-n-2\), multiplied by \(2^n\).
The task is to analyze the series and determine its convergence or divergence and, if possible, calculate its sum.](https://content.bartleby.com/qna-images/question/7949c58c-a906-41a4-97ee-5d0bfe78cf99/1fcc4d28-8e76-405b-b1ec-a83b312c9c7f/9kebe84_processed.jpeg)
Transcribed Image Text:**Instructions:**
Determine whether the series converges or diverges. If it converges, find the sum.
**Mathematical Expression:**
\[
\sum_{n=1}^{\infty} \left( \frac{6}{n^2 + 7n + 12} + 4^{-n-2} \cdot 2^n \right)
\]
**Explanation:**
This series contains two components:
1. **Fractional Term:** \(\frac{6}{n^2 + 7n + 12}\)
- This part represents a rational function where the numerator is constant and the denominator is a quadratic expression in \(n\).
2. **Exponential Term:** \(4^{-n-2} \cdot 2^n\)
- This part includes an exponential expression with a base of 4 raised to the power of \(-n-2\), multiplied by \(2^n\).
The task is to analyze the series and determine its convergence or divergence and, if possible, calculate its sum.
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