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.

(1 + x⁴)⁻¹

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Recall the Taylor series for the function \( \frac{1}{1 - x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \cdots \), which is valid for \( |x| < 1 \).

Notice that the given function \( (1 + x^4)^{-1} \) can be rewritten as \( \frac{1}{1 - (-x^4)} \). This means we can use the geometric series formula by substituting \( -x^4 \) in place of \( x \).

Substitute \( -x^4 \) into the geometric series to get the Taylor series: \[ \frac{1}{1 + x^4} = \sum_{n=0}^{\infty} (-x^4)^n = \sum_{n=0}^{\infty} (-1)^n x^{4n} = 1 - x^4 + x^8 - x^{12} + \cdots \]

Identify the first four nonzero terms from the series expansion: these are the terms corresponding to \( n = 0, 1, 2, 3 \), which are \( 1, -x^4, x^8, -x^{12} \).

Write the final Taylor series approximation centered at 0 using these four terms: \[ 1 - x^4 + x^8 - x^{12} \]. This is the first four nonzero terms of the Taylor series for \( (1 + x^4)^{-1} \).

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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 using polynomials, making complex functions easier to analyze and manipulate.

Manipulating Known Series

Using known Taylor series expansions, such as for (1 - x)⁻¹, allows us to substitute expressions and find series for more complex functions. By replacing variables and adjusting terms, we can derive new series without computing derivatives from scratch.

Identifying Nonzero Terms

When expanding a function into a Taylor series, it is important to recognize which terms are nonzero to write the first few significant terms. This involves understanding the powers of x that appear and simplifying the series accordingly.

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.(1 + x⁴)⁻¹