Power series from the geometric series Use the geometric series a Σₖ ₌ ₀ ∞ (x)ᵏ = 1/(1 - x), for |x| < 1, to determine the Maclaurin series and the interval of convergence for the following functions.


ƒ(x) = ln (1 - 4x)

Verified step by step guidance

1. Recall the geometric series formula: \(\sum_{k=0}^{\infty} x^k = \frac{1}{1 - x}\) for \(|x| < 1\). This will be the foundation for finding the Maclaurin series of \(f(x) = \ln(1 - 4x)\).
2. Recognize that the derivative of \(f(x)\) is \(f'(x) = \frac{d}{dx} \ln(1 - 4x) = \frac{-4}{1 - 4x}\). This derivative can be expressed using the geometric series by rewriting \(\frac{1}{1 - 4x}\) as a power series.
3. Express \(f'(x)\) as a power series: \(f'(x) = -4 \sum_{k=0}^{\infty} (4x)^k = -4 \sum_{k=0}^{\infty} 4^k x^k = - \sum_{k=0}^{\infty} 4^{k+1} x^k\) for \(|4x| < 1\), which simplifies to \(|x| < \frac{1}{4}\).
4. Integrate the power series term-by-term to find \(f(x)\): \(f(x) = \int f'(x) \, dx = - \sum_{k=0}^{\infty} 4^{k+1} \int x^k \, dx = - \sum_{k=0}^{\infty} 4^{k+1} \frac{x^{k+1}}{k+1} + C\).
5. Determine the constant of integration \(C\) by evaluating \(f(0) = \ln(1 - 0) = 0\), which implies \(C = 0\). Thus, the Maclaurin series for \(f(x)\) is \(- \sum_{k=0}^{\infty} \frac{4^{k+1}}{k+1} x^{k+1}\), valid for \(|x| < \frac{1}{4}\).

Key Concepts

Geometric Series and Its Sum Formula

The geometric series is an infinite sum of the form Σ x^k from k=0 to ∞, which converges to 1/(1 - x) when |x| < 1. This formula is fundamental for expressing functions as power series and serves as a starting point for deriving more complex series expansions.

Maclaurin Series Expansion

A Maclaurin series is a Taylor series centered at zero, representing a function as an infinite sum of its derivatives at 0 multiplied by powers of x. It allows approximation of functions near zero and is essential for expressing ln(1 - 4x) as a power series.

Interval of Convergence

The interval of convergence defines the set of x-values for which a power series converges. Determining this interval involves analyzing the radius of convergence, often using the ratio or root test, and is crucial to ensure the validity of the series representation.