Question

Use the definition to find the Taylor series (centered at \(c\) ) for the function. $$ f(x)=e^{x}, \quad c=1 $$

Short answer

The Taylor series for the function \(f(x) = e^{x}\) centered at \(c = 1\) is \[f(x)=\sum_{n=0}^{\infty} \frac{e}{n!}(x-1)^{n}\]

Step-by-step solution

Understand the Taylor series

The Taylor series of a function \(f(x)\) about \(x=c\) is given by the formula: \[ f(x)=\sum_{n=0}^{\infty} \frac{f^{n}(c)}{n!}(x-c)^{n} \]where \(f^{n}(c)\) is the nth derivative of the function evaluated at \(c\).

Compute the nth derivative of the function

For the function \(f(x) = e^{x}\), the nth derivative \(\(f^{n}(x)\)\) is \(e^{x}\), because the derivative of \(e^{x}\) is \(e^{x}\) itself.

Evaluate at \(x=c\)

To get the nth derivative at \(x = c = 1\), plug \(x = 1\) into the nth derivative calculated in Step 2: \[f^{n}(1) = e^{1} = e\] for all \(n\).

Substitute into the Taylor series formula

Now substitute \(f^{n}(1) = e\) and \(c = 1\) into the Taylor series formula:\[f(x)=\sum_{n=0}^{\infty} \frac{e}{n!}(x-1)^{n}\].

Key concepts

Taylor Series Expansion

The concept of a Taylor series expansion is a cornerstone in calculus, particularly useful for approximating complex functions with polynomials. At its heart, the Taylor series represents an infinite sum of terms calculated from the derivatives of a function at a single point.

The general expression for a Taylor series expansion of a function f(x) around a point c is given by:
\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x - c)^{n} \]
where f⁽ⁿ⁾(c) denotes the nth derivative of the function evaluated at point c, and n! stands for n factorial, the product of all positive integers up to n. Each term of this series is a polynomial and the sum of these polynomials approaches the actual function as n increases. This series is particularly powerful because it allows us to evaluate functions that might otherwise be difficult to compute, using a polynomial form.

Exponential Function

The exponential function, denoted as e^x, is among the most important in mathematics due to its unique properties and frequent occurrence in various fields such as economics, physics, and engineering. Its defining characteristic is that the rate of growth is directly proportional to its current value, which intuitively translates to the fact that its derivative is the function itself.

Mathematically, this is expressed as:
\( \frac{d}{dx}e^{x} = e^{x} \)
Because of this property, the derivatives at all orders of the exponential function are equal to the function, which simplifies its Taylor series expansion. When solving problems involving the exponential function, consider it a building block, capable of being manipulated algebraically to suit a variety of scenarios.

Derivatives in Calculus

Derivatives are a fundamental concept in calculus, representing the rate at which a quantity changes as the input changes. In the context of Taylor series, the role of derivatives becomes especially prominent as they form the coefficients of the polynomial approximation of the function.

To obtain the Taylor series, one must calculate the derivatives of the function at a point up to the desired order. For the function e^x, since the derivative at every order is the function itself, it significantly eases the process of finding these coefficients. Derivatives enable us to understand the behavior of functions and predict their future values, which has practical applications in nearly every quantitative field.

Mathematical Series

Mathematical series are sums of terms that follow a specific pattern, often represented as sequences. A series can be finite or infinite, and the Taylor series is an example of an infinite series used to represent functions. In calculus and other areas of mathematics, series are used to simplify complex operations, solve differential equations, and model real-world phenomena.

When dealing with series, particularly infinite ones like the Taylor series, it's important to consider the concept of convergence, which determines whether the sum of the infinite sequence approaches a finite value. This is crucial because it assures us that the series can be used meaningfully in calculations and provides an accurate representation of the function in question.