Problem 48 - Calculus Early Transcendentals

Question

Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series. $$f(x)=\frac{1}{1-x^{4}}$$

Step-by-Step Solution

Answer

Answer: The power series representation centered at 0 for the function $$f(x)=\frac{1}{1-x^{4}}$$ is $$f(x) = \sum_{n=0}^{\infty}(-1)^n x^{4n}$$. The interval of convergence for this power series is \((-1, 1)\).

Step 1: Recall that the geometric series sum formula is given by: $$\sum_{n=0}^{\infty}ar^n = \frac{a}{1-r}$$ where a is the first term and r is the common ratio, and the series converges when \(|r| < 1\). #Step 2: Manipulate the given function to match the geometric series sum formula#

We want to find a power series representation for the given function: $$f(x)=\frac{1}{1-x^{4}}$$ Now, we compare the given function with the geometric series sum formula: $$f(x) = \frac{1}{1-(-x^{4})}$$ Here, we can see that \(a = 1\) and the common ratio \(r = -x^4\). #Step 3: Write the power series representation using the common ratio#

Step 2: Now we write the power series representation using the sum notation and the geometric series formula: $$f(x) = \sum_{n=0}^{\infty}ar^n = \sum_{n=0}^{\infty}(1)(-x^4)^n = \sum_{n=0}^{\infty}(-1)^n x^{4n}$$ #Step 4: Find the interval of convergence for the power series representation#

Recall that the geometric series converges when \(|r| < 1\), where r is the common ratio. In our case, r is \(-x^4\). We want to find the range of x-values for which the series converges: $$|-x^4| < 1 \Rightarrow x^4 < 1$$ Taking the fourth root of both sides, we get: $$|x| < 1$$ Therefore, the interval of convergence for the given power series representation is \((-1, 1)\). So, the power series representation of the given function centered at 0 is: $$f(x) = \sum_{n=0}^{\infty}(-1)^n x^{4n}$$ with the interval of convergence \((-1, 1)\).

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a constant, known as the common ratio. A classic formula for the sum of an infinite geometric series is given by:

• \( \sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} \)

Here, \( a \) represents the first term of the series and \( r \) is the common ratio. The formula provides a means to find the sum only if the absolute value of the common ratio \(|r|\) is less than one.

To use the geometric series in solving problems, it is crucial to identify both the first term and the common ratio in the given context. For instance, in the function \(\frac{1}{1-x^4}\), we recognized it in the format \(\frac{1}{1-r}\), identifying the common ratio as \(-x^4\). By applying the sum formula, this function can be expressed as a power series, which is very useful in calculus.

Interval of Convergence

The interval of convergence is the set of x-values for which the power series converges to a finite sum. In the context of a geometric series, convergence occurs when the absolute value of the common ratio is less than one. This means the series will represent a meaningful result within this interval.

For example, given the power series representation of the function \(f(x) = \sum_{n=0}^{\infty}(-1)^n x^{4n}\), the common ratio is \(-x^4\). To find the interval of convergence, we solve the inequality:

• \(|x^4| < 1\)

Solving this inequality involves taking the fourth root of both sides, which yields \(|x| < 1\). Therefore, the interval of convergence for this power series is \((-1, 1)\), meaning the series is guaranteed to converge for any \(x\) value within this interval.

Convergence Criteria

Understanding convergence criteria is vital for determining if a power series represents a function effectively. Specifically for geometric series, convergence depends on the common ratio \(r\).

For convergence:

• The absolute value \(|r|\) must be less than 1.
• If \(|r| \geq 1\), the series diverges, meaning it does not sum to a finite value.

In our problem, the common ratio \(r = -x^4\). Therefore, the series is convergent within the interval where \(|-x^4| < 1\) simplifies to \(|x| < 1\). This criterion ensures that the series sums to a valid function within the specified range.

These criteria help us decide where our series representation is valid and thus dictate the interval over which a function can be accurately described by its power series form.