Question

Taylor series and interval of convergence

a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.

f(x) = tan ⁻¹ (x/2), a = 0

Accepted Answer

Recall that the Taylor series of a function $$f(x)$$ centered at $$a is $$given by the formula:

$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n,$$

where $$f^{(n)}(a$$)$$ is the n$-th derivative of f$ evaluated at x = a$. Since the center is a = 0$, this is a Maclaurin series. We need to find the first four nonzero terms of the series for f(x$$) = \$$tan^{-1}$$\left(\frac{x}{2}ight)$$, so we will compute the derivatives of f(x$$)$$ at x=0$ up to the order that gives four nonzero terms. Start by computing the first derivative:

f$$'(x) = \frac{d}{dx} 	an^{-1}\left(\frac{x}{2}ight) = \frac{1}{1 + \left(\frac{x}{2}ight)^2} \cdot \frac{1}{2} = \frac{1}{2 \left(1 + \frac{x^2}{4}ight)} = \frac{1}{2 + \frac{x^2}{2}}.$$

Evaluate f$$'(0)$$ to get the coefficient for the linear term. Next, find the higher order derivatives f$$''(x)$$, f$$^{(3)}$$(x)$$, and $$f^{(4)}(x$$)$$, and evaluate each at x=0$. Use these values to write the terms

$$\frac{f^{(n)}(0)}{n!} x^n$$

for n=0$$,1,2,3$$ (or until you have four nonzero terms). Finally, write out the Taylor series expansion up to the fourth nonzero term by summing these terms:

f(x$$) \approx f(0) + \frac{f'(0)}{1!} x + \frac{f''(0)}{2!} x^2 + \frac{f^{(3)}(0)}{3!} x^3 + \cdots$$

This will give you the first four nonzero terms of the Maclaurin series for f(x$$) = \$$tan^{-1}$$\left(\frac{x}{2}ight)$$.

Taylor series and interval of convergence

a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.

f(x) = tan ⁻¹ (x/2), a = 0

Verified video answer for a similar problem:

Key Concepts

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Interval of Convergence

Taylor series and interval of convergencea. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.f(x) = tan ⁻¹ (x/2), a = 0

Verified video answer for a similar problem:

Key Concepts

Taylor and Maclaurin Series

Derivatives of Inverse Trigonometric Functions

Interval of Convergence

Taylor series and interval of convergence

a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.

f(x) = tan ⁻¹ (x/2), a = 0