The Taylor series is a fundamental concept in mathematics, particularly in calculus, that allows us to represent a function as an infinite sum of terms that are expressed in terms of the values of the function’s derivatives at a single point. For the function (f(x) = \frac{1}{1+x}), the Taylor series expansion around (x = 0) is a classic example that illustrates the power and utility of this mathematical tool.
Introduction to Taylor Series
Before diving into the specifics of the Taylor series for (f(x) = \frac{1}{1+x}), let’s briefly review what a Taylor series is. The Taylor series of a function (f(x)) about (x = a) is given by:
[f(x) = f(a) + \frac{f’(a)}{1!}(x-a) + \frac{f”(a)}{2!}(x-a)^2 + \frac{f”‘(a)}{3!}(x-a)^3 + \cdots]
This series representation provides a way to approximate functions near a point (a) using an infinite polynomial, and it’s particularly useful when dealing with functions that are complicated or difficult to compute directly.
The Function (f(x) = \frac{1}{1+x})
For the specific function (f(x) = \frac{1}{1+x}), we’re interested in finding its Taylor series expansion around (x = 0). This means we’ll be calculating (f(0)), (f’(0)), (f”(0)), and so on, to plug into the Taylor series formula.
Calculating Derivatives
To find the Taylor series, we first calculate the derivatives of (f(x) = \frac{1}{1+x}) evaluated at (x = 0):
First Derivative: (f’(x) = \frac{-1}{(1+x)^2})
- (f’(0) = \frac{-1}{(1+0)^2} = -1)
Second Derivative: (f”(x) = \frac{2}{(1+x)^3})
- (f”(0) = \frac{2}{(1+0)^3} = 2)
Third Derivative: (f”‘(x) = \frac{-6}{(1+x)^4})
- (f”’(0) = \frac{-6}{(1+0)^4} = -6)
Fourth Derivative: (f”“(x) = \frac{24}{(1+x)^5})
- (f”“(0) = \frac{24}{(1+0)^5} = 24)
And so on. Notice the pattern in the derivatives and their values at (x = 0): the sign alternates, and the factorial of the derivative order appears in the denominator of the derivative’s value at (x = 0).
Constructing the Taylor Series
Given these derivatives, we construct the Taylor series:
[f(x) = f(0) + f’(0)x + \frac{f”(0)}{2!}x^2 + \frac{f”‘(0)}{3!}x^3 + \frac{f”“(0)}{4!}x^4 + \cdots]
Plugging in the calculated values:
[f(x) = 1 - x + \frac{2}{2!}x^2 - \frac{6}{3!}x^3 + \frac{24}{4!}x^4 - \cdots]
Simplifying:
[f(x) = 1 - x + x^2 - x^3 + x^4 - \cdots]
This is the Taylor series expansion of (f(x) = \frac{1}{1+x}) around (x = 0).
Formula Guide
For practical purposes, the Taylor series of (\frac{1}{1+x}) around (x = 0) can be represented by the formula:
[\frac{1}{1+x} = \sum_{n=0}^{\infty} (-1)^n x^n]
This formula is valid for (-1 < x < 1), which is the interval of convergence for this series.
Applications and Conclusion
The Taylor series expansion of (f(x) = \frac{1}{1+x}) is a foundational tool in calculus and analysis. It not only helps in approximating the function near (x = 0) but also serves as a building block for more complex Taylor series expansions and theoretical developments in mathematics and physics.
In conclusion, understanding and applying Taylor series, such as the one for (\frac{1}{1+x}), is crucial for advancing in calculus, differential equations, and other areas of mathematics and science. The formula guide provided here offers a concise way to express and work with this particular series, illustrating the power of Taylor series in representing functions in a manageable and insightful form.
What is the interval of convergence for the Taylor series of f(x) = \frac{1}{1+x}?
+The interval of convergence for the Taylor series of f(x) = \frac{1}{1+x} is -1 < x < 1.
How do I determine the number of terms to use in the Taylor series expansion for an accurate approximation?
+The number of terms depends on the desired accuracy and the value of x. Generally, more terms are required for x values closer to the boundaries of the interval of convergence.
In summary, the Taylor series expansion of (f(x) = \frac{1}{1+x}) around (x = 0) offers a valuable method for approximating and understanding the behavior of this function within its interval of convergence. By following the guidelines and considering the applications and limitations of this series, one can effectively utilize this mathematical tool in a variety of contexts.
