Problem 17 - Calculus: One and Several Variables

Question

Find a series expansion for the expression. $$\frac{x}{x+x^{2}} \quad \text { for }|x|<1$$

Step-by-Step Solution

Answer

The series expansion for the given expression \(\frac{x}{x+x^2}\) for |x|<1 is \(\sum_{n=0}^{\infty} (-x)^n\).

Step 1: Rewrite the expression into a more convenient form

First, we need to rewrite the given expression in the form of a geometric series. We will factor the denominator: \[ \frac{x}{x+x^2} = \frac{x}{x(1+x)} \] Now we can rewrite the expression as: \[ \frac{1}{1+x} \]

Step 2: Identify the geometric series

Next, we can recognize that the expression \(\frac{1}{1+x}\) is a geometric series with the common ratio r = -x. The general formula for a geometric series is: \[ \sum_{n=0}^{\infty} ar^n \] where a is the first term and r is the common ratio. In our case, the first term (a) is 1 and the common ratio (r) is -x. So our geometric series can be written as: \[ \sum_{n=0}^{\infty} 1(-x)^n \]

Step 3: Rewrite the series expansion for the given domain

Now we can rewrite our series expansion as follows, for |x|<1: \[ \frac{1}{1+x} = \sum_{n=0}^{\infty} (-x)^n \] This is the series expansion of the given expression \(\frac{x}{x+x^2}\) for |x|<1.

Key Concepts

Geometric Series

A geometric series is a sequence of terms where each term is derived from the previous one by multiplying with a constant called the common ratio. For example:

• If you have a series with starting term 2 and you multiply each subsequent term by -1/2, the series becomes 2, -1, 0.5, -0.25, and so on.
• The general form of a geometric series is written as \( a + ar + ar^2 + ar^3 + ... \).

This form can also be expressed using a summation notation:\[\sum_{n=0}^{fty} ar^n\] where \( a \) is the first term and \( r \) is the common ratio. These series are simple and powerful because, when the absolute value of the common ratio is less than 1, the series converges, allowing the sum of an infinite number of terms.

Common Ratio

The common ratio in a geometric series is the factor by which we multiply each term to get to the next term. It's denoted by \( r \). For the series \( a, ar, ar^2, ... \), the common ratio is \( r \).

• In the example from the solution, our expression \( \frac{1}{1+x} \) has a common ratio of -x.
• This means each term in the series is generated by multiplying the previous term by \(-x\).

Identifying the common ratio is essential, as it influences convergence and dictates the behavior of the series. It's what makes the series geometric, and without a consistent ratio, the series could not maintain its structured progression.

Infinite Series

An infinite series, like its name suggests, is a series that continues indefinitely. It's the sum of an infinite number of terms. When dealing with an infinite series:

• We often investigate whether the series converges to a particular value as more terms are added.
• The notable aspect is that even though there are infinitely many terms, their sum can sometimes be finite.

In the provided problem, the series expansion of \( \frac{1}{1+x} \) using the series \( \sum_{n=0}^{\infty} (-x)^n \) illustrates how we express functions as sums over an infinite series within a certain interval.

Radius of Convergence

The radius of convergence is a crucial concept when examining infinite series, especially when it comes to power series. It tells us within which values of \( x \) the series converges:

• For a power series like \( \sum_{n=0}^{\infty} c_nx^n \), there's a radius \( R \).
• The series will only converge if \(|x|

For the series \( \sum_{n=0}^{\infty} (-x)^n \), the radius of convergence is 1, meaning that the series converges for \(|x|<1\). This defines a safe zone, ensuring that if \( x \) falls within this interval, the series can be summed to produce a meaningful result. Outside of this, the series could diverge or become undefined, emphasizing why understanding the radius of convergence is so important in series math problems.