Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series


(1 + x)⁻² = 1 − 2x + 3x² − 4x³ + ⋯, for −1 < x < 1.


(1 + 4x)⁻²

Verified step by step guidance

1. Recall the given Maclaurin series for the function \((1 + x)^{-2}\): \[ (1 + x)^{-2} = 1 - 2x + 3x^{2} - 4x^{3} + \cdots \]
2. To find the Maclaurin series for \((1 + 4x)^{-2}\), use substitution by replacing \(x\) with \$4x$ in the original series. This gives: \[ (1 + 4x)^{-2} = 1 - 2(4x) + 3(4x)^{2} - 4(4x)^{3} + \cdots \]
3. Simplify each term by calculating the powers and coefficients: - The linear term: \(-2 \times 4x = -8x\) - The quadratic term: \$3 \times (4x)^2 = 3 \times 16x^2 = 48x^2\( - The cubic term: \)-4 \times (4x)^3 = -4 \times 64x^3 = -256x^3$
4. Write out the first four nonzero terms explicitly: \[ (1 + 4x)^{-2} = 1 - 8x + 48x^{2} - 256x^{3} + \cdots \]
5. Confirm the interval of convergence remains valid by considering the substitution: since the original series converges for \(|x| < 1\), the new series converges for \(|4x| < 1\), or equivalently \(|x| < \frac{1}{4}\).

Key Concepts

Maclaurin Series

A Maclaurin series is a special case of the Taylor series expanded at x = 0. It represents a function as an infinite sum of terms involving powers of x and derivatives evaluated at zero. Understanding this allows approximation of functions near zero using polynomials.

Binomial Series Expansion

The binomial series generalizes the binomial theorem to any real exponent, expressing (1 + x)^n as an infinite power series. For negative integer exponents, it produces alternating terms with coefficients derived from binomial coefficients, useful for expanding functions like (1 + x)^-2.

Substitution and Factoring in Power Series

Substitution involves replacing x with another expression inside a known power series to find expansions of related functions. Factoring helps simplify expressions before expansion. Together, they enable finding series for functions like (1 + 4x)^-2 by adapting the known series for (1 + x)^-2.