The Taylor series expansion of $f(x) = \ln(1 + 2x)$ about $x = 0$ is $\sum_{n=1}^\infty (-1)^{n-1} \frac{(2x)^n}{n}$. The radius of convergence $R$ of this series is:

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### 1. Strategy & Theoretical Principles By the ratio test for power series $\sum a_n x^n$: $$R = \lim_{n \to \infty} \left| \frac{c_n}{c_{n+1}} \right|$$ Alternatively, for $f(z) = \ln(1 + 2z)$, the function is analytic everywhere except at the branch singularity where $1 + 2z = 0 \implies z = -1/2$. The distance from the expansion center $z_0 = 0$ to the nearest singularity is the radius of convergence $R$. ### 2. Analytical Formulation & Calculation Nearest singularity: $$1 + 2x = 0 \implies x = -\frac{1}{2}$$ $$R = |0 - (-1/2)| = \frac{1}{2} = 0.5$$ Or by the ratio test: $$\left| \frac{u_{n+1}}{u_n} \right| = \left| \frac{(2x)^{n+1}}{n+1} \frac{n}{(2x)^n} \right| = |2x| \left(\frac{n}{n+1}\right) \to |2x|$$ Convergence requires $|2x| < 1 \implies |x| < 1/2 \implies R = 0.5$. ### 3. Distractor Forensics - Candidate confusing with standard $\ln(1+x)$ reports $R = 1.0$. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Radius of convergence is determined by the distance to the nearest singularity in the complex plane. - **Standard Reference:** Kreyszig, E., *Advanced Engineering Mathematics*, Chapter 15.

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