A useful substitution Replace x with x−1 in the series ln (1+x) = ∑ₖ₌₁∞ ((−1)ᵏ⁺¹ xᵏ)/k to obtain a power series for ln x centered at x = 1. What is the interval of convergence for the new power series?

Verified step by step guidance

1. Start with the given power series for \( \ln(1+x) \): \[ \ln(1+x) = \sum_{k=1}^{\infty} (-1)^{k+1} \frac{x^k}{k} \] which is centered at \( x=0 \) and converges for \( -1 < x \leq 1 \).
2. Make the substitution \( x \to x - 1 \) to express \( \ln x \) in terms of \( x - 1 \): \[ \ln x = \ln(1 + (x-1)) = \sum_{k=1}^{\infty} (-1)^{k+1} \frac{(x-1)^k}{k} \] This gives a power series for \( \ln x \) centered at \( x = 1 \).
3. Determine the interval of convergence for the new series by considering the original interval \( -1 < x \leq 1 \) for \( \ln(1+x) \). Since the substitution is \( x = t + 1 \), the original variable \( x \) in the series corresponds to \( t = x - 1 \). The original interval \( -1 < x \leq 1 \) becomes \( -1 < t \leq 1 \), so for \( t = x - 1 \), the interval is \( -1 < x - 1 \leq 1 \).
4. Solve the inequalities for \( x \): \[ -1 < x - 1 \leq 1 \implies 0 < x \leq 2 \] This means the power series for \( \ln x \) centered at \( x=1 \) converges for \( x \) in the interval \( (0, 2] \).
5. Check the endpoints for convergence: - At \( x = 0 \), the series does not converge because \( \ln x \) is not defined at zero. - At \( x = 2 \), substitute into the series and verify convergence using the alternating series test or other convergence tests. Thus, the interval of convergence for the new power series is \( (0, 2] \).

Key Concepts

Power Series and Centering

A power series is an infinite sum of terms involving powers of (x - a), where 'a' is the center. Changing variables or substituting x with (x - 1) shifts the center of the series, allowing us to express functions like ln(x) as a series centered at x = 1 instead of 0.

Interval of Convergence

The interval of convergence is the set of x-values for which a power series converges. It depends on the radius of convergence and the center. After substitution, the interval shifts accordingly, and determining it involves analyzing the original series' convergence and the effect of the substitution.

Substitution in Power Series

Substitution involves replacing the variable in a known power series with an expression like (x - 1). This transforms the series to represent a related function centered at a new point. Careful substitution helps derive new series expansions and their convergence properties.