Maclaurin Series
Find Taylor series at x = 0 for the functions in Exercises 63–70.
cos (x³/√5)

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Recall that the Maclaurin series is a Taylor series expansion of a function at \(x = 0\). It can be written as \(f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n\).

Identify the function to expand: \(f(x) = \cos\left(\frac{x^3}{\sqrt{5}}\right)\). Notice that the argument of cosine is \(\frac{x^3}{\sqrt{5}}\).

Recall the Maclaurin series expansion for \(\cos z\) is \(\cos z = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n}}{(2n)!}\), where \(z\) is any expression.

Substitute \(z = \frac{x^3}{\sqrt{5}}\) into the cosine series to get \(\cos\left(\frac{x^3}{\sqrt{5}}\right) = \sum_{n=0}^{\infty} (-1)^n \frac{\left(\frac{x^3}{\sqrt{5}}\right)^{2n}}{(2n)!}\).

Simplify the powers and write the series explicitly as \(\sum_{n=0}^{\infty} (-1)^n \frac{x^{6n}}{(2n)! (\sqrt{5})^{2n}}\). This is the Maclaurin series for the given function.

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주요 개념

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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 its derivatives at zero, multiplied by powers of x and divided by factorial terms. This series helps approximate functions near zero.

Taylor Series Expansion

The Taylor series expresses a function as an infinite sum of terms calculated from the function's derivatives at a specific point. For a function f(x), the series at x = a is given by f(a) plus derivatives evaluated at a, scaled by powers of (x - a). When a = 0, it becomes the Maclaurin series.

Series Expansion of Composite Functions

When dealing with functions like cos(x³/√5), the series expansion involves substituting the inner function into the known series of the outer function. This requires understanding how to replace variables in standard series and simplify powers and coefficients accordingly.

Maclaurin SeriesFind Taylor series at x = 0 for the functions in Exercises 63–70.cos (x³/√5)