Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.


ln (1 + x²)

Verified step by step guidance

1. Recall the Taylor series expansion for \( \ln(1+x) \) centered at 0, which is given by: \[ \ln(1+x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \] This is the series you will manipulate to find the series for \( \ln(1+x^2) \).
2. Substitute \( x^2 \) in place of \( x \) in the series for \( \ln(1+x) \). This gives: \[ \ln(1+x^2) = x^2 - \frac{(x^2)^2}{2} + \frac{(x^2)^3}{3} - \frac{(x^2)^4}{4} + \cdots \] which simplifies to: \[ \ln(1+x^2) = x^2 - \frac{x^4}{2} + \frac{x^6}{3} - \frac{x^8}{4} + \cdots \]
3. Identify the first four nonzero terms from the series after substitution. These terms correspond to the powers of \( x \) with nonzero coefficients in the expansion.
4. Write out explicitly the first four nonzero terms of the Taylor series for \( \ln(1+x^2) \) centered at 0, using the simplified powers and coefficients from the previous step.
5. Verify that the series is centered at 0 and that the terms are ordered by increasing powers of \( x \). This confirms the correctness of the Taylor series expansion for \( \ln(1+x^2) \).

Key Concepts

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives at a single point, usually centered at zero (Maclaurin series). It approximates functions locally and is useful for expressing complex functions as polynomials.

Manipulating Known Series

To find the Taylor series of a new function, you can use known series expansions and apply algebraic operations like substitution, multiplication, or composition. For example, substituting x² into the series for ln(1 + x) helps find the series for ln(1 + x²).

Identifying Nonzero Terms

When expanding a series, it is important to identify and list the first few nonzero terms, as some terms may vanish due to the function's structure or substitution. This ensures the approximation captures the function's behavior accurately near the center.