Taylor Series Of E^{-17x} At X=0: Step-by-Step

by Sebastian Müller 47 views

Hey guys! Today, we're diving deep into the fascinating world of Taylor series, specifically focusing on how to find the Taylor series of the function e-17x at x = 0. We'll also figure out the general expression for the nth term in this Taylor series. So, buckle up and let's get started!

Understanding Taylor Series: The Foundation

Before we jump into the specifics of e-17x, let's take a moment to understand what a Taylor series actually is. At its core, a Taylor series is a representation of a function as an infinite sum of terms, each of which is calculated from the function's derivatives at a single point. Think of it as a way to approximate a function using polynomials, which are often much easier to work with. Taylor series are incredibly powerful tools in mathematics, physics, and engineering, allowing us to analyze and approximate complex functions with simpler polynomial expressions. They form the bedrock of many numerical methods and are used extensively in solving differential equations, evaluating integrals, and approximating function values. The ability to express a function as a Taylor series unlocks a world of possibilities for analysis and computation, making it a fundamental concept in advanced mathematics.

The general form of a Taylor series for a function f(x) centered at a is given by:

f(x) = Σn=0∞ [f(n)(a) / n!] (x - a)n

Where:

  • f(n)(a) is the nth derivative of f evaluated at a.
  • n! is the factorial of n.
  • a is the point around which the series is centered.

When a = 0, the Taylor series is also known as a Maclaurin series. This is the specific case we'll be dealing with today, as we're finding the Taylor series at x = 0. This Maclaurin series is a special case of the Taylor series and is often simpler to compute because we're evaluating the function and its derivatives at zero. This simplification makes the Maclaurin series particularly useful for approximating functions near the origin and for various analytical calculations. The Maclaurin series provides a powerful way to represent functions as power series, which have numerous applications in calculus, differential equations, and complex analysis. Understanding the Maclaurin series is crucial for anyone working with advanced mathematical concepts and their applications.

The Power of Substitution: A Clever Trick

Now, instead of directly computing derivatives (which can get messy!), we're going to use a clever trick called substitution. This technique leverages the known Taylor series of a simpler function to find the Taylor series of a more complex one. This method is particularly effective when dealing with composite functions, where direct differentiation can become cumbersome. Substitution allows us to bypass the need for repeated differentiation by utilizing the series representation of a known function. This approach not only simplifies the process but also provides a deeper understanding of the relationship between different functions and their series representations. By recognizing patterns and applying appropriate substitutions, we can efficiently derive Taylor series for a wide range of functions, making it a valuable tool in mathematical analysis.

We know the Maclaurin series for the exponential function ex is:

ex = Σn=0∞ xn / n! = 1 + x + x2/2! + x3/3! + ...

This is a fundamental series that's worth memorizing! The Maclaurin series for ex is a cornerstone in calculus and analysis, appearing in numerous applications across various fields. Its simple and elegant form makes it easy to manipulate and use as a building block for finding series representations of more complex functions. The convergence properties of this series are well-understood, making it a reliable tool for approximations and analytical calculations. Mastering the Maclaurin series for ex is essential for anyone working with Taylor series and their applications.

To find the Taylor series for e-17x, we can simply substitute -17x for x in the Maclaurin series for ex. This is where the magic happens! By making this substitution, we're essentially transforming the known series into one that represents our target function. This technique is a powerful illustration of how understanding the series representations of basic functions can be leveraged to find series for more complex ones. The substitution method is not only efficient but also provides insights into the structural relationships between functions and their Taylor series. It highlights the versatility of Taylor series and their ability to be manipulated to solve a variety of problems.

Finding the Taylor Series of e^{-17x}

Let's do the substitution! Replacing x with -17x in the series for ex, we get:

e-17x = Σn=0∞ (-17x)n / n!

This is the Taylor series for e-17x at x = 0. Pretty neat, huh? This series representation allows us to understand the behavior of the function e-17x near the origin. It provides a polynomial approximation that can be used for calculations and analysis, particularly in situations where evaluating the exponential function directly might be difficult or computationally expensive. The Taylor series representation also reveals the function's derivatives at x = 0, which can be useful for solving differential equations and other problems. Overall, this result provides a powerful tool for working with the function e-17x.

The General Expression for the nth Term

Now, let's nail down the general expression for the nth term in this Taylor series. Looking at our result:

e-17x = Σn=0∞ (-17x)n / n!

We can see that the nth term is given by:

an = (-17)n xn / n!

This is the formula that generates each term in the series. This general expression is crucial for understanding the pattern and structure of the Taylor series. It allows us to calculate any term in the series without having to compute derivatives or perform repeated substitutions. The nth term formula encapsulates the essence of the Taylor series, showing how the function's behavior is captured by an infinite sum of power terms. This expression is not only useful for calculations but also for theoretical analysis, providing insights into the convergence and properties of the series.

Putting It All Together: A Final Look

So, to recap, we've successfully found the Taylor series of e-17x at x = 0 using the substitution method. We started with the known Maclaurin series for ex, substituted -17x for x, and arrived at the series for our target function. We then identified the general expression for the nth term. This entire process showcases the elegance and efficiency of using Taylor series and substitution techniques to represent and analyze functions. The combination of these methods allows us to tackle complex problems with relative ease, making them indispensable tools in mathematics and related fields.

This process highlights the following key takeaways:

  1. Substitution is a powerful technique for finding Taylor series.
  2. Knowing the Maclaurin series for ex is super helpful.
  3. The general expression for the nth term allows us to understand the series' structure.

Understanding Taylor series and techniques like substitution are fundamental in calculus and analysis. They provide a way to approximate functions, solve differential equations, and explore the behavior of functions in various contexts. Mastering these concepts opens doors to a deeper understanding of mathematics and its applications in science and engineering.

Why This Matters: Real-World Applications

You might be wondering,