Problem 12 - Single Variable Calculus

Question

Use the power series representations of functions established in this section to find the Taylor series of \(f\) at the given value of \(c .\) Then find the radius of convergence of the series. \(f(x)=\frac{1}{1+x}, \quad c=-2\)

Step-by-Step Solution

Answer

The Taylor series for the function \(f(x) = \frac{1}{1+x}\) centered at \(c = -2\) is given by: \[f(x) = \sum_{n=0}^{\infty} (-1)^n(x + 2)^n.\] The radius of convergence for this series is R = 1.

Step 1: Recall that the geometric series is given as \[\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n,\] with a radius of convergence R = 1. Since our function is \(f(x)= \frac{1}{1+x}\), we can rewrite this as \[\frac{1}{1-(-x)}.\] By using the geometric series formula, we find the representation of our function: \[f(x) = \sum_{n=0}^{\infty} (-x)^n.\] #Step 2: Substitute x with x + 2 to get the Taylor series at c = -2#

To obtain the power series around the point \(c = -2\), we need to make the substitution \(x = x + 2\). Plugging the expression into our original power series, we get: \[f(x) = \sum_{n=0}^{\infty} (-(x + 2))^n.\] #Step 3: Simplify the power series#

Step 2: We simplify the given power series by expanding the (-1)^n(x + 2)^n term: \[f(x) = \sum_{n=0}^{\infty} (-1)^n(x + 2)^n.\] This is the Taylor series for the function \(f(x) = \frac{1}{1+x}\) centered at \(c = -2\). #Step 4: Find the radius of convergence#

Since the power series representation we found came from the geometric series \(\frac{1}{1-x}\), the radius of convergence remains the same (R = 1). So, the radius of convergence for the Taylor series of \(f(x) = \frac{1}{1+x}\) centered at \(c = -2\) is R = 1.

Key Concepts

Power Series

A power series is a type of infinite series that represents a function as an infinite sum of terms, each term being a multiple of a power of the variable. The general formula for a power series centered at a point, say \(c\), is given by: \[ \sum_{n=0}^{\infty} a_n(x - c)^n. \]
Here, each \(a_n\) is a coefficient that can be determined through various methods, and \(x\) is the variable.
Power series are useful for approximating functions, particularly those that may be difficult to express in a closed form.

• They can be differentiated and integrated term-by-term within their interval of convergence, making them flexible tools in calculus.
• Power series can express a wide range of functions, particularly if these functions are difficult to compute normally.

For example, the exponential function \(e^x\), trigonometric functions like \( an^{-1}(x)\), and logarithmic functions can all be represented as power series.

Radius of Convergence

The radius of convergence of a power series is the distance within which the series converges to the function it represents.
It is crucial to determine this radius to ensure that the power series gives accurate results within specific limits. For a power series centered at \(c\), the radius of convergence, \(R\), might be found using methods such as the Ratio Test or the Root Test.

Consider a power series \[ \sum_{n=0}^{\infty} a_n(x - c)^n. \]

• If \(R > 0\), then for any \(x\) such that \(|x - c| < R\), the power series converges.
• When \(R = 0\), the series converges only at \(x = c\).
• If \(R = \infty\), the series converges for all real numbers \(x\).

In our example, the radius of convergence for the series \(\frac{1}{1+x}\) is \(R = 1\), which means the series accurately represents the function for \(-1 < x < 1\).

Geometric Series

A geometric series is a specific type of series where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
The general form of a geometric series is: \[ a + ar + ar^2 + ar^3 + \cdots = \sum_{n=0}^{\infty} ar^n. \] Here, \(a\) is the first term, and \(r\) is the common ratio. A simple and classic example of a geometric series is the sum \(\frac{1}{1-x} \), which is \(\sum_{n=0}^{\infty} x^n\) as long as the absolute value of \(x\) is less than 1.

• The geometric series formula becomes foundational in simplifying more complex functions into a series form using calculus.
• It's often used as a "building block" for understanding other kinds of series.

For the exercise example, by identifying the function \(\frac{1}{1+x}\) as a type involving a geometric series, students can comfortably express the function as an infinite series.

Calculus

Calculus is the branch of mathematics that studies continuous change. It is used to understand phenomena in fields ranging from physics and engineering to economics and biology. Calculus focuses on the concepts of derivatives and integrals.
These are used to model patterns of change. Derivatives give the rate at which one quantity changes with respect to another, while integrals evaluate the accumulation of quantities.

• In terms of power series and Taylor series, calculus allows for the differentiation and integration of series term by term when convergent.
• Finding the nth derivative is a crucial step in developing Taylor series, which represent functions as infinite sums.

Calculus aids in finding the behavior of functions near points of interest, like \(c = -2\) in our problem. Understanding foundational calculus concepts ensures you can effectively use Taylor series to approximate functions. This is a powerful technique as it lets us evaluate functions that are not easy to work with directly. By using calculus within the framework of power and Taylor series, we gain a deep understanding of functional behavior around specific points.