Question

Sketch a graph of \(f(x)=\frac{1}{1-x^{2}}\) and the corresponding partial sums \(S_{N}(x)=\sum_{n=0}^{N} x^{2 n}\) fo \(N=2,4,6\) on the interval \((-1,1)\)

Short answer

The function \( f(x) = \frac{1}{1-x^2} \) has vertical asymptotes at \( x = 1 \) and \( x = -1 \); the partial sums \( S_N(x) \) converge to \( f(x) \) as \( N \) increases.

Step-by-step solution

Understand the Function

The function given is \( f(x) = \frac{1}{1-x^2} \). This function exhibits vertical asymptotes at \( x = 1 \) and \( x = -1 \), since the denominator becomes zero at these points. The domain of \( f(x) \) for real numbers is \( x \in (-1, 1) \).

Examine Partial Sums of Series

The series representing the function is \( S_{N}(x) = \sum_{n=0}^{N} x^{2n} \). For different values of \( N \), this partial sum approximates the function \( \frac{1}{1-x^2} \) within the interval \((-1,1)\).

Calculate Partial Sums for Given N

Compute the partial sums for \( N = 2, 4, 6 \):- For \( N = 2 \), \( S_2(x) = 1 + x^2 + x^4 \).- For \( N = 4 \), \( S_4(x) = 1 + x^2 + x^4 + x^6 + x^8 \).- For \( N = 6 \), \( S_6(x) = 1 + x^2 + x^4 + x^6 + x^8 + x^{10} + x^{12} \).

Plot the Function \( f(x) \)

Plot \( f(x) = \frac{1}{1-x^2} \) over the interval \((-1, 1)\). Notice the curve tends to infinity as it approaches \( x = 1 \) and \( x = -1 \), showing vertical asymptotes.

Plot Partial Sums \( S_N(x) \)

For each \( N = 2, 4, 6 \), plot \( S_N(x) \) over the interval \((-1, 1)\). These should approximate \( f(x) \) more closely as \( N \) increases, demonstrating the convergence of the series.

Compare Graphs for Analysis

Compare the plotted graphs of \( f(x) \) and \( S_N(x) \). Notice that as \( N \) gets larger, the partial sums \( S_N(x) \) closely follow \( f(x) \) within the interval \((-1, 1)\), illustrating how the series converges towards the function.

Key concepts

Function Graphing

Graphing a function is a visual way to understand its behavior through a plot of its curve on a coordinate plane. When graphing the function \( f(x) = \frac{1}{1-x^2} \), you will notice some distinct characteristics within the interval \((-1, 1)\). The curve represents the relationship between \( x \) and \( f(x) \), showing how \( f(x) \) changes as \( x \) varies.

To start, you need to mark the interval between the vertical asymptotes \( x = -1 \) and \( x = 1 \) on the x-axis. Then, calculate and plot several specific points by choosing values for \( x \) within \((-1, 1)\). This will aid in sketching a smooth curve that approaches infinity as it nears the boundaries of the interval.

Using appropriate software or a graphing calculator can make this process straightforward and precise. Remember that each plot point provides essential clues on how the function behaves across its domain.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. They are essential in determining the behavior of functions, especially rational functions like \( f(x) = \frac{1}{1-x^2} \).

• Vertical Asymptotes: These occur where the function is undefined and its values tend to infinity. For our function, the denominator \( 1-x^2 \) becomes zero at \( x = 1 \) and \( x = -1 \), creating vertical asymptotes at these points.
• Horizontal Asymptotes: Although not present in this function within \((-1, 1)\), they occur when \( x \) tends towards infinity or negative infinity, indicating the end behavior of the function.

Knowing the location of asymptotes helps when sketching the graph. It tells us where the graph blows up and how it behaves near these critical values. Keep these in mind when plotting, as they drastically affect the graph's appearance.

Convergence

Convergence is a key concept when dealing with power series. It refers to how a series approaches a particular value as more terms are added.

In this exercise, the power series \( S_N(x) \) aims to approximate \( f(x) = \frac{1}{1-x^2} \) within \((-1,1)\). The partial sums like \( S_2(x) = 1 + x^2 + x^4 \) are approximations that refine the more terms you include. As \( N \) increases, the series becomes a closer approximation of the function within the given domain.

When graphing, you'll observe that for higher values of \( N \), the partial sums overlay more closely with the function's curve, demonstrating strong convergence. This is a powerful method in mathematics for handling infinite series, allowing approximations of functions that might otherwise be difficult to work with directly.

Partial Sums

Partial sums are essentially the sum of a specified number of terms in a series. For the function \( S_N(x) = \sum_{n=0}^{N} x^{2n} \), each partial sum represents another step in approximating \( \frac{1}{1-x^2} \).

Let's break it down for different \( N \):

• For \( N = 2 \), the series is \( S_2(x) = 1 + x^2 + x^4 \).
• For \( N = 4 \), it's \( S_4(x) = 1 + x^2 + x^4 + x^6 + x^8 \).
• For \( N = 6 \), we have \( S_6(x) = 1 + x^2 + x^4 + x^6 + x^8 + x^{10} + x^{12} \).

Each step involves adding more terms to the series, thereby making \( S_N(x) \) an increasingly accurate approach of the initial function over its domain. When graphed alongside \( f(x) \), these partial sums demonstrate how close each approximation is, enabling a visual representation of the series' convergence.