Maclaurin Series for y xe^(-x): A Comprehensive Guide

Understanding the Maclaurin series for a function is crucial for various applications in mathematics and engineering. In this article, we will explore the Maclaurin series representation of the function y xe^-x.

Introduction to the Maclaurin Series

The Maclaurin series is a power series representation of a function around the point x 0. It is an expansion of the function f(x) in the form of an infinite sum of terms involving the derivatives of f evaluated at x 0.

Mathematically, the Maclaurin series for a function f(x) is given by:

f(x) f(0) f'(0)x frac{f''(0)}{2!}x^2 frac{f'''(0)}{3!}x^3 cdots

Maclaurin Series for e^x

Let's start with the Maclaurin series for the exponential function e^x.

e^x 1 x frac{x^2}{2!} frac{x^3}{3!} frac{x^4}{4!} cdots

Deriving the Maclaurin Series for e^(-x)

To find the Maclaurin series for e^(-x), we will substitute -x for x in the series for e^x.

e^(-x) 1 - x frac{x^2}{2!} - frac{x^3}{3!} frac{x^4}{4!} - cdots

Deriving the Maclaurin Series for y xe^(-x)

Now, to find the Maclaurin series for the function y xe^(-x), we multiply the Maclaurin series for e^(-x) by x.

y xe^(-x) x[1 - x frac{x^2}{2!} - frac{x^3}{3!} frac{x^4}{4!} - cdots]

Expanding and simplifying:

y x - x^2 frac{x^3}{2!} - frac{x^4}{3!} frac{x^5}{4!} - cdots

Verification and Application

Let's verify the series expansion by calculating a few terms manually:

First term: x Second term: -x^2 Third term: frac{x^3}{2!} Fourth term: -frac{x^4}{3!} Fifth term: frac{x^5}{4!}

Thus, the Maclaurin series for the function y xe^(-x) is:

y x - x^2 frac{x^3}{2!} - frac{x^4}{3!} frac{x^5}{4!} - cdots

Conclusion

In conclusion, understanding the Maclaurin series for y xe^(-x) is essential for various analytical and computational tasks. This series can be used to approximate values and solve differential equations involving the given function.

If you have any questions or need further assistance, feel free to leave a comment below. Happy learning!