Problem 42 - Single Variable Calculus

Question

(a) find the power series representation for the function; (b) write the first three partial sums \(P_{1}, P_{2}\), and \(P_{3} ;\) and \((\mathrm{c})\) plot the graphs of \(f\) and \(P_{1}, P_{2}\), and \(P_{3}\) using a viewing window that includes the interval of convergence of the power series. \(f(x)=\frac{1}{\sqrt{9-x}}\)

Step-by-Step Solution

Answer

In summary, the power series representation for the given function \(f(x)=\frac{1}{\sqrt{9-x}}\) is \(f(x) = \sum_{n=0}^{\infty} \frac{1}{9^n}x^n\). The first three partial sums are \(P_1 = 1\), \(P_2 = 1 + \frac{1}{9}x\), and \(P_3 = 1 + \frac{1}{9}x + \frac{1}{81}x^2\). The interval of convergence is \(-9 < x < 9\). Plot the function and its partial sums within this interval using a graphing tool or software.

Step 1: Find the power series representation of the function

First, let's rewrite the given function in a more recognizable form that resembles the geometric series formula: \(A + Ar + Ar^{2} + ...\), where \(A\) is the first term, and \(r\) is the common ratio. Notice that: \(\frac{1}{\sqrt{9-x}} = \frac{1}{\sqrt{9(1-\frac{x}{9})}} = \frac{1}{{\sqrt{9}\times\sqrt{1-\frac{x}{9}}}} = \frac{1}{3\sqrt{1-\frac{x}{9}}}\) Now this form is identical to the geometric series: \(1 + \frac{x}{9} + \frac{x^2}{81} + ...\) In general, the power series representation is: \(f(x) = \sum_{n=0}^{\infty} \frac{1}{9^n}x^n\)

Step 2: Find the first three partial sums of the power series

We can find the first three partial sums, \(P_1\), \(P_2\), and \(P_3\), by taking the sum of the first one, first two, and first three terms of the power series, respectively. \(P_1 = f_1(x) = \frac{1}{1} =1\) \(P_2 = f_2(x) = \frac{1}{1} + \frac{1}{9}x = 1 + \frac{1}{9}x\) \(P_3 = f_3(x) = \frac{1}{1} + \frac{1}{9}x + \frac{1}{81}x^2 = 1 + \frac{1}{9}x + \frac{1}{81}x^2\)

Step 3: Plot the graphs of the function and its partial sums

To plot the graphs of the function and its partial sums over the interval of convergence, we need to first identify this interval. For a geometric series, the interval of convergence is determined by the absolute value of the common ratio: \(|r| < 1\). In this case, the common ratio is \(\frac{x}{9}\), so we can write: \(|\frac{x}{9}| < 1 \Rightarrow -1 <\frac{x}{9} < 1 \Rightarrow -9 < x < 9\) (Notice that the endpoints of the interval are not included in the interval of convergence, due to the inequality being strict.) Now, we can use a graphing tool or software to graph the function \(f(x) = \frac{1}{\sqrt{9-x}}\) and the partial sums: \[ P_1 = 1, P_2 = 1 + \frac{1}{9}x, P_3 = 1 + \frac{1}{9}x + \frac{1}{81}x^2 \] across the interval \(-9 < x < 9\).

Key Concepts

Partial Sums

Partial sums are important when dealing with power series because they provide approximations to the entire series. When we calculate partial sums, we essentially stop after a certain number of terms, capturing a snapshot of the series up to that point.
For example, if you have a power series like this one:

• The first partial sum, \(P_1\), is just the first term of the series.
• The second partial sum, \(P_2\), adds the second term to the first.
• The third partial sum, \(P_3\), includes the third term as well, forming a more refined approximation.

In this exercise, finding partial sums for the series helped us understand the behavior of the function \(f(x) = \frac{1}{\sqrt{9-x}}\) by looking at simpler versions that progressively include more detail as we add each term.

Interval of Convergence

The interval of convergence tells us where a power series is valid or useful. It's a range of values for which the series converges to a definite value, meaning the partial sums get closer to a single number as you add more terms.
In this problem, we determined the interval of convergence by analyzing the condition for convergence of the geometric series. We derived that the series converges when \(|\frac{x}{9}| < 1\). After solving this inequality, we found that:

• The interval is \(-9 < x < 9\).
• This interval excludes the endpoints, as the inequality is strict.
• The series accurately represents the function only within this range.

Understanding where the power series converges helps in identifying the applicable range when plotting or interpreting series behavior.

Geometric Series

Geometric series are a type of series with each term being a constant multiple of the previous one. This constant is known as the common ratio, denoted usually by \(r\). The series appears as:\(A + Ar + Ar^2 + Ar^3 + \dots\)where \(A\) is the first term.
In this case, we transformed \(f(x)\) into a form that matches the geometric series pattern:

• The starting term \(A\) and common ratio \(r\) set how each next term is constructed.
• For our function \(f(x) = \frac{1}{3\sqrt{1-\frac{x}{9}}}\), using properties of geometric series allowed for expressing \(f(x)\) as a series where \(r = \frac{x}{9}\).
• This series gives us powerful tools to approximate and analyze functions using just sums of simple terms.

Function Graphing

Graphing functions and their partial sums gives a visual insight into how well the series approximates the function. With each added term, the partial sum should more closely align with the function. For the given function \(f(x) = \frac{1}{\sqrt{9-x}}\), and its partial sums:

• \(P_1 = 1\) captures just the most basic form.
• \(P_2 = 1 + \frac{1}{9}x\) includes linear behavior.
• \(P_3 = 1 + \frac{1}{9}x + \frac{1}{81}x^2\) adds a quadratic component, improving the approximation further.

When graphed over the interval of convergence \(-9 < x < 9\), these plots illustrate convergence to the function as the asterisk (*) in the interval signifies points outside which the series does not validly represent the function. These visual tools are vital for checking how well the series and function match up in practice.