To solve this problem, we need to expand the function $f(x) = 3x \ln(1 + 2x)$ as a power series and find specific coefficients $c_0, c_1, c_2, c_3, c_4$, as well as the radius of convergence $R$ of the series.

Step 1: Expand $\ln(1 + 2x)$ as a Power Series

We can start by using the Taylor series expansion for $\ln(1 + u)$: $\ln(1 + u) = u - \frac{u^2}{2} + \frac{u^3}{3} - \frac{u^4}{4} + \cdots$ Setting $u = 2x$, we get: $\ln(1 + 2x) = 2x - \frac{(2x)^2}{2} + \frac{(2x)^3}{3} - \frac{(2x)^4}{4} + \cdots$ Simplifying each term, we find: $\ln(1 + 2x) = 2x - 2x^2 + \frac{8x^3}{3} - 2x^4 + \cdots$

Step 2: Multiply by $3x$ to Find $f(x)$

Now, multiply this series by $3x$ to obtain $f(x) = 3x \ln(1 + 2x)$: $f(x) = 3x \left(2x - 2x^2 + \frac{8x^3}{3} - 2x^4 + \cdots \right)$ Expanding each term, we get: $f(x) = 6x^2 - 6x^3 + 8x^4 - 6x^5 + \cdots$

Step 3: Identify Coefficients $c_0, c_1, c_2, c_3, c_4$

Now we can identify the coefficients of $f(x)$ in the power series form $f(x) = \sum_{n=0}^{\infty} c_n x^n$:

• $c_0 = 0$ (no constant term)
• $c_1 = 0$ (no $x$ term)
• $c_2 = 6$
• $c_3 = -6$
• $c_4 = 8$

Step 4: Find the Radius of Convergence $R$

The radius of convergence $R$ of the series $\sum c_n x^n$ can be found using the formula: $\frac{1}{R} = \lim_{n \to \infty} \left| \frac{c_{n+1}}{c_n} \right|$ Since the function $f(x) = 3x \ln(1 + 2x)$ involves $\ln(1 + 2x)$, the radius of convergence for this series is determined by the interval of convergence for $\ln(1 + 2x)$, which converges for $|2x| < 1$, or $|x| < \frac{1}{2}$.

Thus, $R = \frac{1}{2}$.

Final Answers

• $c_0 = 0$
• $c_1 = 0$
• $c_2 = 6$
• $c_3 = -6$
• $c_4 = 8$
• Radius of convergence $R = \frac{1}{2}$

Would you like further clarification on any part of this solution?

Related Questions

1. How do you derive the Taylor series for $\ln(1 + u)$?
2. What is the general formula for finding the radius of convergence of a power series?
3. Why do certain functions, like logarithmic functions, have restricted radii of convergence?
4. How would the solution change if the function were $3x^2 \ln(1 + 2x)$?
5. What is the significance of each coefficient in the power series expansion?

Tip

When dealing with logarithmic functions in series expansions, remember that they often converge within a limited interval due to their inherent nature.