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)=3ˣ, a=0

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

Identify the function: \(f(x) = 3^x\). To find the Maclaurin series, we need to compute the derivatives of \(f(x)\) and evaluate them at \(x=0\).

Calculate the first four derivatives of \(f(x)\): - \(f(x) = 3^x\) - \(f'(x) = 3^x \ln(3)\) - \(f''(x) = 3^x (\ln(3))^2\) - \(f^{(3)}(x) = 3^x (\ln(3))^3\) - \(f^{(4)}(x) = 3^x (\ln(3))^4\)

Evaluate each derivative at \(x=0\): - \(f(0) = 3^0 = 1\) - \(f'(0) = 3^0 \ln(3) = \ln(3)\) - \(f''(0) = 3^0 (\ln(3))^2 = (\ln(3))^2\) - \(f^{(3)}(0) = 3^0 (\ln(3))^3 = (\ln(3))^3\)

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

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주요 개념

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Taylor and Maclaurin Series

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

To find the Taylor series of f(x) = 3^x, you need to compute its derivatives at the center point. The derivative of an exponential function a^x is a^x times the natural logarithm of a. This pattern repeats for higher-order derivatives, which is essential for determining the series coefficients.

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 3^x, the series converges for all real numbers, meaning the interval of convergence is (-∞, ∞). Understanding this ensures the series accurately represents the function within this domain.

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)=3ˣ, a=0