Evaluate as power series and determine the radius of convergence. x2 -dx 5+ 2x
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Must be simplified but remain in exact form (no rounding)
![**Problem Statement:**
Evaluate the following integral as a power series and determine the radius of convergence:
\[ \int \frac{x^2}{5 + 2x} \, dx \]
**Instructions for Solution:**
1. **Identify the Series Expression:**
Decompose the integrand \(\frac{x^2}{5 + 2x}\) into a form suitable for series expansion.
2. **Expand into Power Series:**
Use the geometric series and other relevant series expansions to express \(\frac{1}{5 + 2x}\) as a power series.
3. **Integrate Term by Term:**
Once the power series for the integrand is determined, perform term-by-term integration on the series.
4. **Determine the Radius of Convergence:**
Apply the ratio test or other convergence tests to find the interval of convergence for the series.
5. **Summarize the Result:**
Write the resulting power series and clearly state the radius of convergence.
**Note:** There are no graphs or diagrams included in this problem statement. The main focus is on the algebraic manipulation and series expansion.
**Solution Outline:**
1. **Rewrite Integrand for Series Expansion:**
\[ \frac{x^2}{5 + 2x} = x^2 \cdot \frac{1}{5} \cdot \frac{1}{1 + \frac{2x}{5}} \]
2. **Use Geometric Series Expansion:**
\[ \frac{1}{1 + \frac{2x}{5}} = \sum_{n=0}^{\infty} \left(-\frac{2x}{5}\right)^n \]
3. **Write the Power Series:**
\[ \frac{x^2}{5 + 2x} = \frac{x^2}{5} \sum_{n=0}^{\infty} \left(-\frac{2x}{5}\right)^n = \sum_{n=0}^{\infty} \frac{x^2}{5} \left(-\frac{2x}{5}\right)^n \]
4. **Simplify and Integrate Term by Term:**
\[ \int \sum_{n=0}^{\infty} \frac{x^2}{5} \left(-\frac{2x}{5}\right)^](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5643f6a4-a7a4-4b03-93d4-ff1f1b43abe4%2F672acab7-6b69-4a45-a2b6-f5a9aeafaf7b%2Fj6y21ag_processed.png&w=3840&q=75)
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