Problem 23 - Essential Calculus

Question

Use the power series $$\frac{1}{1+x}=\sum_{n=0}^{\infty}(-1)^{n} x^{n}$$ to determine a power series, centered at 0 , for the function. Identify the interval of convergence. $$ h(x)=\frac{1}{4 x^{2}+1} $$

Step-by-Step Solution

Answer

The power series for the function \( h(x) = \frac{1}{1 + 4x^2} \) centered at 0 is \( h(x) = \sum_{n=0}^{\infty}(-1)^{n} 4^{n}x^{2n} \) , and the interval of convergence is \( -\frac{1}{2} < x < \frac{1}{2} \)

Step 1: Express Given Function as Geometric Series

Write the equation \( h(x) = \frac{1}{4x^2 + 1} \) as \( h(x) = \frac{1}{1 + 4x^2} \) so the expression conforms to the form \( \frac{1}{1+x} \).

Step 2: Substitute Into Power Series

Substitute \( 4x^2 \) for \( x \) in the power series \( \sum_{n=0}^{\infty}(-1)^{n} x^{n} \) yielding \( h(x) = \sum_{n=0}^{\infty}(-1)^{n} (4x^2)^{n} \).

Step 3: Simplify Power Series

Simplify the series \( h(x) = \sum_{n=0}^{\infty}(-1)^{n} (4x^2)^{n} \) to get \( h(x) = \sum_{n=0}^{\infty}(-1)^{n} 4^{n}x^{2n} \). This is the power series for \( h(x) = \frac{1}{1 + 4x^2} \) centered at 0.

Step 4: Find Interval of Convergence

Remember that a geometric series \( \sum_{n=1}^{\infty}ar^{n -1} \) converges only if \( |r| < 1 \). Here, \( r = 4x^{2} \). Hence, for the series to converge, \( |4x^2| < 1 \). Solve this inequality for \( x \) to find the interval of convergence, which is \( -\frac{1}{2} < x < \frac{1}{2} \).

Key Concepts

Geometric Series

To understand how we transform functions into power series, it's important to start with the geometric series. A geometric series is a series of the form:

• \( \sum_{n=0}^{\infty} ar^n \)

Here, \( a \) is the first term and \( r \) is the common ratio. In the simplest form where \( a = 1 \), it becomes:

• \( \sum_{n=0}^{\infty} x^n = \frac{1}{1-x} \)

This series converges under the condition that \( |x| < 1 \). We use this idea to express more complex functions as a series. Any function that can be represented in the form \( \frac{1}{1+u} \) can be expressed as a geometric series. For example, with \( u = 4x^2 \) in the function \( h(x) = \frac{1}{1+4x^2} \), we substitute \( -4x^2 \) into the series formula, resulting in:

• \( h(x) = \sum_{n=0}^{\infty} (-1)^n (4x^2)^n \)

This expanded series helps in understanding and calculating function behaviors over specific intervals.

Interval of Convergence

The interval of convergence is crucial for determining where a power series is valid. For a geometric series \( \sum_{n=0}^{\infty} ar^n \) to converge, the absolute value of the common ratio \( r \) must be less than 1.

• \( |r| < 1 \)

In our exercise, \( r = 4x^2 \). Hence, we set up the inequality:

• \( |4x^2| < 1 \)

Solving this inequality gives us the interval of convergence. Simplifying:

• \(|4| \times |x^2| < 1\)
• \(|x^2| < \frac{1}{4} \)
• \(-\frac{1}{2} < x < \frac{1}{2} \)

This interval \( -\frac{1}{2} < x < \frac{1}{2} \) tells us the range of \( x \) values for which the series \( h(x) = \sum_{n=0}^{\infty} (-1)^n 4^n x^{2n} \) converges, ensuring the series effectively represents the function within this interval.

Convergence Criteria

The convergence criteria stem from understanding where a series represents a function properly. For geometric series, the convergence condition \( |r| < 1 \) guides us. It finds application, not just in pure geometrical settings, but also in power series expansions. More specifically:

• The series \( \sum_{n=0}^{\infty} ar^n \) only holds for values of \( x \) that make \( |ar^n| \) decrease indefinitely as \( n \) increases.

For our power series, which substitutes \( r = 4x^2 \), convergence occurs only if this product remains less than 1:

• \( |4x^2| < 1 \)

This criterion ensures the series doesn't diverge or blow up for values of \( x \) outside the specified interval. Thus, solving:

• -\(\frac{1}{2} < x < \frac{1}{2}\)

It forms the basis not only for understanding the series' interval of reliability but also for directing where mathematical transformations are accurate and simplifying complex functions into infinite series representations.