Taylor series and interval of convergence

b. Write the power series using summation notation.

f(x) = e⁻ˣ, a=0

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Recall the Taylor series expansion of a function \( f(x) \) about \( a = 0 \) (also called the Maclaurin series) is given by: \[ \displaystyle f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \] where \( f^{(n)}(0) \) is the \( n \)-th derivative of \( f \) evaluated at 0.

Identify the function given: \( f(x) = e^{-x} \). The derivatives of \( e^{-x} \) follow a pattern because the derivative of \( e^{-x} \) is \( -e^{-x} \). Specifically, \[ \begin{cases} f(x) = e^{-x} \\ f'(x) = -e^{-x} \\ f''(x) = e^{-x} \\ f^{(3)}(x) = -e^{-x} \\ \vdots \end{cases} \]

Evaluate the \( n \)-th derivative at \( x=0 \): \[ \displaystyle f^{(n)}(0) = (-1)^n e^{0} = (-1)^n \] since \( e^{0} = 1 \).

Substitute \( f^{(n)}(0) = (-1)^n \) into the Taylor series formula to write the power series in summation notation: \[ \displaystyle e^{-x} = \sum_{n=0}^{\infty} \frac{(-1)^n}{n!} x^n \]

This summation notation represents the power series expansion of \( e^{-x} \) centered at \( a=0 \).

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Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. For a function f(x) centered at a, the series is given by summing (f⁽ⁿ⁾(a)/n!) * (x - a)ⁿ, where n ranges from 0 to infinity. This allows approximation of functions using polynomials.

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Power Series and Summation Notation

A power series is an infinite series of the form Σ cₙ(x - a)ⁿ, where cₙ are coefficients and a is the center. Summation notation (Σ) concisely expresses this infinite sum, making it easier to write and manipulate series. Writing the Taylor series in summation form clarifies the pattern of coefficients and powers.

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Interval of Convergence

The interval of convergence is the set of x-values for which a power series converges to a finite value. It is found by applying convergence tests like the ratio test. Understanding this interval is crucial because the Taylor series only accurately represents the function within this range.

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