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) = e²ˣ, a = 0

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Recall the definition of the Taylor series of a function \(f(x)\) centered at \(a\): \[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n,\] where \(f^{(n)}(a)\) is the \(n\)-th derivative of \(f\) evaluated at \(x = a\).

Identify the function and the center: here, \(f(x) = e^{2x}\) and \(a = 0\), so we are looking for the Maclaurin series (Taylor series at 0).

Compute the derivatives of \(f(x)\) and evaluate them at \(x=0\): - \(f(x) = e^{2x}\) - \(f'(x) = 2e^{2x}\) - \(f''(x) = 4e^{2x}\) - \(f^{(3)}(x) = 8e^{2x}\) Evaluate each at \(x=0\): - \(f(0) = e^0 = 1\) - \(f'(0) = 2e^0 = 2\) - \(f''(0) = 4e^0 = 4\) - \(f^{(3)}(0) = 8e^0 = 8\)

Write the first four nonzero terms of the Taylor series using the formula: \[f(x) \approx f(0) + \frac{f'(0)}{1!} x + \frac{f''(0)}{2!} x^2 + \frac{f^{(3)}(0)}{3!} x^3,\] and substitute the values found in the previous step.

Simplify each term by dividing the derivative values by the factorials and write the series explicitly up to the \(x^3\) term.

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주요 개념

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Taylor and Maclaurin Series Definition

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. When centered at zero, it is called a Maclaurin series. Each term involves the nth derivative evaluated at the center, multiplied by (x - a)^n and divided by n!.

Derivatives of Exponential Functions

The exponential function e^(kx) has derivatives that are proportional to itself, specifically the nth derivative is k^n * e^(kx). This property simplifies finding terms in the Taylor series since each derivative at the center can be easily computed.

Interval of Convergence

The interval of convergence is the set of x-values for which the Taylor series converges to the function. For exponential functions like e^(2x), the series converges for all real x, meaning the interval of convergence is (-∞, ∞).

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) = e²ˣ, a = 0