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.

sinh x²

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Recall the Taylor series expansion for the hyperbolic sine function centered at 0: \(\sinh x = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!} = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \cdots\).

To find the Taylor series for \(\sinh(x^2)\), substitute \(x^2\) in place of \(x\) in the series for \(\sinh x\). This gives: \(\sinh(x^2) = \sum_{n=0}^{\infty} \frac{(x^2)^{2n+1}}{(2n+1)!} = \sum_{n=0}^{\infty} \frac{x^{4n+2}}{(2n+1)!}\).

Write out the first four nonzero terms explicitly by plugging in \(n=0,1,2,3\) into the series: \(\frac{x^{2}}{1!} + \frac{x^{6}}{3!} + \frac{x^{10}}{5!} + \frac{x^{14}}{7!}\).

Simplify the factorials in the denominators where possible: \(1! = 1\), \(3! = 6\), \(5! = 120\), \(7! = 5040\).

Express the first four nonzero terms of the Taylor series for \(\sinh(x^2)\) as: \(x^{2} + \frac{x^{6}}{6} + \frac{x^{10}}{120} + \frac{x^{14}}{5040} + \cdots\).

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주요 개념

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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 compute.

Hyperbolic Sine Function (sinh x)

The hyperbolic sine function, sinh x, is defined as (e^x - e^(-x))/2. Its Taylor series expansion at zero includes only odd powers of x with alternating signs, which helps in constructing series for related functions like sinh(x²).

Substitution in Series Expansions

Substitution involves replacing the variable in a known Taylor series with another expression, such as x², to find the series of composite functions. This technique allows leveraging existing expansions to find new series efficiently.

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.sinh x²